{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Blue Emphasis" -1 256 "Times" 1 12 0 0 255 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 258 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Maroon Emphasis" -1 259 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 260 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Purple Emphasis" -1 261 " Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis" -1 262 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 283 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 286 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 287 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 288 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 289 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 290 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 291 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 292 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 293 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 294 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emp hasis" -1 295 "Times" 1 12 255 255 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "P urple Emphasis" -1 296 "Times" 1 12 255 0 255 1 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 297 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 298 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 299 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 300 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 301 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 302 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 303 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 304 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 305 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 306 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 307 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 308 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 309 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet \+ Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet It em" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Normal" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 260 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 257 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 39 "Continuous and discontinuous func tions " }{TEXT 263 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 18 "Version: 22.3.2 007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 40 "The concept of continuity o f a function " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "The limit of a function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG 6#%\"xG" }{TEXT -1 4 " as " }{TEXT 264 1 "x" }{TEXT -1 21 " approaches a number " }{TEXT 279 1 "a" }{TEXT -1 52 " can often by found simply \+ by calculating the value " }{XPPEDIT 18 0 "f(a);" "6#-%\"fG6#%\"aG" } {TEXT -1 17 " of the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG " }{TEXT -1 4 " at " }{XPPEDIT 18 0 "x = a;" "6#/%\"xG%\"aG" }{TEXT -1 10 ", so that " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 257 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(f(x),x = a) = f(a);" "6#/-%&Lim itG6$-%\"fG6#%\"xG/F*%\"aG-F(6#%\"aG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "When this happens we s ay that the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 4 " is " }{TEXT 261 10 "continuous" }{TEXT -1 4 " at " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Roughly, a function f is contin uous at " }{XPPEDIT 18 0 "x = a;" "6#/%\"xG%\"aG" }{TEXT -1 29 " provi ded that the values of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 21 " change gradually as " }{TEXT 277 1 "x" }{TEXT -1 29 " in creases through the value " }{XPPEDIT 18 0 "x = a;" "6#/%\"xG%\"aG" } {TEXT -1 51 " with no sudden jump in value. Continuity of f at " } {XPPEDIT 18 0 "x = a;" "6#/%\"xG%\"aG" }{TEXT -1 30 " corresponds to t he graph of " }{XPPEDIT 18 0 "y = f(x);" "6#/%\"yG-%\"fG6#%\"xG" } {TEXT -1 24 " going through the point" }{XPPEDIT 18 0 "``(a,f(a));" "6 #-%!G6$%\"aG-%\"fG6#%\"aG" }{TEXT -1 16 " with no break. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 354 267 267 {PLOTDATA 2 "6.-%'CURVESG6%7S7$$\"3++++++++]!#=$\"3/LLLLLL 3s!#<7$$\"35mmmT:(z@&F*$\"3-,yv)ycdM(F-7$$\"3jLLe9ui2aF*$\"31wWJZqFnuF -7$$\"3Anm;z_\"4i&F*$\"3&\\pHCdBhg(F-7$$\"3$pmmT&phNeF*$\"3))y=Z8[C[xF -7$$\"35LLe*=)H\\gF*$\"3C`0'3Zb?*yF-7$$\"3;nm\"z/3uC'F*$\"3uK\\OM5^F!) F-7$$\"37++DJ$RDX'F*$\"3uDzrM(Q*p\")F-7$$\"3'fm;zR'okmF*$\"3cp'zxcz&>$ )F-7$$\"3I++D1J:woF*$\"3aU'*p3\"H6Z)F-7$$\"3WLLL3En$4(F*$\"30ou]=X_H') F-7$$\"3qmm;/RE&G(F*$\"3e?NLiA;r()F-7$$\"3\")*****\\K]4](F*$\"3nORyu., L*)F-7$$\"3$******\\PAvr(F*$\"3!=$=2$e(4)4*F-7$$\"3`+++v'Hi#zF*$\"3_\" G;Ck\\'f#*F-7$$\"3jmm\"z*ev:\")F*$\"3r/AIe;Y3%*F-7$$\"3kKLL347T$)F*$\" 3Q<&zS`V!)e*F-7$$\"3,LLLLY.K&)F*$\"38`&)3*QCCu*F-7$$\"3?***\\7o7Tv)F*$ \"3K`RtKQiC**F-7$$\"3IKLL$Q*o]*)F*$\"35q`E\"pE)35!#;7$$\"3A++D\"=lj;*F *$\"3nx!yiRQq-\"Fgq7$$\"3]***\\PaR#\\ECjW5Fgq7$$\"3!HLLe9E ge*F*$\"3#=5k86aK1\"Fgq7$$\"3GLLeR\"3Gy*F*$\"3)*40'z!Rf!3\"Fgq7$$\"3cm m;/T1&***F*$\"3c%>gC%eb*4\"Fgq7$$\"3em;zRQb@5F-$\"3^x&*)*f\"Q&>6Fgq7$$ \"3%)**\\(=>Y2/\"F-$\"3(fG)H.>\"Fgq7$$\"3#***\\7y%3T7\"F-$\"3r30pDAQ;7Fgq7$$\"3#****\\P! [hY6F-$\"3%HC([Zr]Q7Fgq7$$\"3ELLLQx$o;\"F-$\"3K3VX5#f'e7Fgq7$$\"3')*** *\\P+V)=\"F-$\"33f-$*4=Y!G\"Fgq7$$\"3im;zpe*z?\"F-$\"3RsBB8\\Z+8Fgq7$$ \"3)*****\\#\\'QH7F-$\"3:LP*>iNEK\"Fgq7$$\"37L$e9S8&\\7F-$\"3(Hvs0%pvV 8Fgq7$$\"3%***\\i?=bq7F-$\"3=q^&eD;hO\"Fgq7$$\"3GLL$3s?6H\"F-$\"3]#zz$ fkD)Q\"Fgq7$$\"3&***\\7`Wl78F-$\"3m[7h6Ot69Fgq7$$\"3emmm'*RRL8F-$\"3#* e\\;1`jM9Fgq7$$\"3_mmTvJga8F-$\"3y*G>>=_$e9Fgq7$$\"3KL$e9tOcP\"F-$\"3/ epEE2<#[\"Fgq7$$\"3'******\\Qk\\R\"F-$\"3je#f'z2K/:Fgq7$$\"3@LL3dg6<9F -$\"3%)fZmo#>+`\"Fgq7$$\"3_mmmw(GpV\"F-$\"3'*)G2wM)G`:Fgq7$$\"3-+]7oK0 e9F-$\"3+CY\"[H&Ry:Fgq7$$\"3-+](=5s#y9F-$\"3B\\E3&)[r-;Fgq7$$\"3++++++ ++:F-$\"3Vmmmmm;H;Fgq-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!Fa[lF`[l-%* THICKNESSG6#\"\"#-F$6$7$7$F`[lF`[l7$$Fe[lFa[lF`[l-Fjz6&F\\[lFa[lFa[lFa [l-F$6%7$7$$\"\"\"Fa[lF`[l7$Fb\\l$\"#6Fa[lF\\\\l-%*LINESTYLEGFd[l-F$6& 7#Fd\\l-%'SYMBOLG6#%'CIRCLEGFiz-%&STYLEG6#%&POINTG-F$6&F[]l-F]]l6#%(DI AMONDGFizF`]l-F$6&F[]l-F]]l6#%&CROSSGFizF`]l-%%TEXTG6%7$Fb\\l$!\"'!\" \"Q&x~=~a6\"-%&COLORG6&F\\[l$Fc\\l!\"#F`[lF`[l-F_^l6%7$$\"#&)F[_l$\"#7 Fa[lQ)(a,f(a))Ff^lFg^l-F_^l6%7$$\"$X\"F[_l$\"#8Fa[lQ)y~=~f(x)Ff^lFiz-% +AXESLABELSG6%Q\"xFf^lQ!Ff^l-%%FONTG6#%(DEFAULTG-%*AXESSTYLEG6#%%NONEG -%%VIEWG6$;F`[lF[\\l;Fb^l$\"# " 0 "" {MPLTEXT 1 0 93 "g := x -> cos(40000*x):\nf := x -> x^2+abs(g(x))/(1000*g(x)):\n'f( x)'=f(x);\nplot(f(x),x=-3..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% \"fG6#%\"xG,&*$)F'\"\"#\"\"\"F,*&#F,\"%+5F,*&-%$absG6#-%$cosG6#,$*&\"& ++%F,F'F,F,F,F4!\"\"F,F," }}{PARA 13 "" 1 "" {GLPLOT2D 339 306 306 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$!\"$\"\"!$\"3c++++++***)!#<7$$!3!***** *\\2<#pGF-$\"3&RbrMi1MB)F-7$$!3#)***\\7bBav#F-$\"3#*fJyY*e8f(F-7$$!36+ +]K3XFEF-$\"3NW\\?xy\\/pF-7$$!3%)****\\F)H')\\#F-$\"3azE([,^TC'F-7$$!3 #****\\i3@/P#F-$\"3KH'Qh7'*yh&F-7$$!3;++Dr^b^AF-$\"3kiUC\"F-$\"3k#z2))Ffp$\"3u$3QQM?xw(Ffp7$$!3S++]7RKvuFfp$\"3 WJ%z'fn/)f&Ffp7$$!3s,+++P'eH'Ffp$\"3-rx*G(**ytRFfp7$$!3q)***\\7*3=+&Ff p$\"33UkrR#4=\\#Ffp7$$!3[)***\\PFcpPFfp$\"3S[[>B.'4V\"Ffp7$$!3;)****\\ 7VQ[#Ffp$\"3ec(4'pmZpg!#>7$$!32)***\\i6:.8Ffp$\"3;'4DB&H?)f\"Fgr7$$!3W b+++v`hH!#?$\"3f1RO/2x35F`s7$$\"3]****\\(QIKH\"Ffp$\"3!))R`^$[Ws:Fgr7$ $\"38****\\7:xWCFfp$\"3_L:L[x!p(eFgr7$$\"3E,++vuY)o$Ffp$\"3a)G89Bz/P\" Ffp7$$\"3!z******4FL(\\Ffp$\"3!)Q%fVC)R$[#Ffp7$$\"3A)****\\d6.B'Ffp$\" 3u&*y:K#y;(QFfp7$$\"3s****\\(o3lW(Ffp$\"3f'H,L;\\]`&Ffp7$$\"35*****\\A ))oz)Ffp$\"38jVJWU_GxFfp7$$\"3e******Hk-,5F-$\"37W&eN\"R0,5F-7$$\"36++ +D-eI6F-$\"3!40h^k6sF\"F-7$$\"3u***\\(=_(zC\"F-$\"3i4ThY@We:F-7$$\"3M+ ++b*=jP\"F-$\"3#GH#*e'QD&*=F-7$$\"3g***\\(3/3(\\\"F-$\"3$e1j-v\\-C#F-7 $$\"33++vB4JB;F-$\"3')yg;b$QTj#F-7$$\"3u*****\\KCnu\"F-$\"3#[qYv'e/_IF -7$$\"3s***\\(=n#f(=F-$\"3^S,7a55=NF-7$$\"3P+++!)RO+?F-$\"3?T9[_gX+SF- 7$$\"30++]_!>w7#F-$\"3*z*4cKGwFXF-7$$\"3O++v)Q?QD#F-$\"3a-_uWjqy]F-7$$ \"3G+++5jypBF-$\"3t;M1br)[h&F-7$$\"3<++]Ujp-DF-$\"3exGw#)*)[iiF-7$$\"3 ++++gEd@EF-$\"3Wv%f;@V;(oF-7$$\"39++v3'>$[FF-$\"39`(R=ngAb(F-7$$\"37++ D6EjpGF-$\"3%[\\\\NK\"zL#)F-7$$\"\"$F*F+-%'COLOURG6&%$RGBG$\"#5!\"\"$F *F*F`[l-%+AXESLABELSG6$Q\"x6\"Q!Fe[l-%%VIEWG6$;F(Fgz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "When the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 30 " is plotted over the inter val " }{XPPEDIT 18 0 "-3<=x" "6#1,$\"\"$!\"\"%\"xG" }{XPPEDIT 18 0 "`` <=3" "6#1%!G\"\"$" }{TEXT -1 41 " the graph appears to be a smooth cur ve. " }}{PARA 0 "" 0 "" {TEXT -1 114 "However, zooming in to a small p art of the graph shows that it consists of many disconnected pieces. T he function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 27 " h as many discontinuities. " }}{PARA 0 "" 0 "" {TEXT -1 46 "Formally, th e discontinuities of the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#% \"xG" }{TEXT -1 9 " are the " }{TEXT 266 1 "x" }{TEXT -1 63 " values ( members of the domain) at which jumps in value occur. " }}{PARA 0 "" 0 "" {TEXT -1 71 "In order to see the discontinuities it is necessary \+ to provide Maple's " }{TEXT 0 4 "plot" }{TEXT -1 27 " procedure with t he option " }{TEXT 262 12 "discont=true" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "d := \+ .0005;\nplot(f(x),x=2-d..2+d,discont=true,thickness=1);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"dG$\"\"&!\"%" }}{PARA 13 "" 1 "" {GLPLOT2D 420 335 335 {PLOTDATA 2 "6%-%'CURVESG627S7$$\"31+++++]**>!#<$\"3#***** **\\-+**RF*7$$\"3MyH%)G,]**>F*$\"3y'3V_w+!**RF*7$$\"3qk&[4C+&**>F*$\"3 7oLb87+**RF*7$$\"3e@I-n.]**>F*$\"3u(>Dxr,!**RF*7$$\"3u\"=LR\\+&**>F*$ \"3eP!R_A-!**RF*7$$\"3'*f,C?1]**>F*$\"3C%yS.t-!**RF*7$$\"3C(*GMP2]**>F *$\"3e*yM')>.!**RF*7$$\"3!))*ffe3]**>F*$\"3AOh_$o.!**RF*7$$\"3uHh*R)4] **>F*$\"3!3\\,]=/!**RF*7$$\"3/,T**36]**>F*$\"3!)R'o[o/!**RF*7$$\"3')y) pvB,&**>F*$\"3=xM/*>0!**RF*7$$\"3)Q\\>3N,&**>F*$\"3]!)*G>l0!**RF*7$$\" 33PF*$\"3o@3xhh+**RF*7$$\"3)fZFjg,&**>F*$\"3qchqtm+**RF*7$$\"3 !4O%pH<]**>F*$\"3+T20nr+**RF*7$$\"3C6LsT=]**>F*$\"3U8\\0:w+**RF*7$$\"3 +pl$\\(>]**>F*$\"3]R_xZ\"3!**RF*7$$\"3urcy(3-&**>F*$\"3))f#f!*f3!**RF* 7$$\"3!f+c!>A]**>F*$\"3s\"*)4S74!**RF*7$$\"3a%\\__L-&**>F*$\"3')y,o)e4 !**RF*7$$\"3ce%QFY-&**>F*$\"3#3;(\\)45!**RF*7$$\"3GW_8%e-&**>F*$\"3]>N 'Re5!**RF*7$$\"3#\\0+3r-&**>F*$\"3=on\\!46!**RF*7$$\"3S\"*y6FG]**>F*$ \"3QSClb:,**RF*7$$\"3'>R#e_H]**>F*$\"3I.dQd?,**RF*7$$\"35ZZ!H3.&**>F*$ \"3!zeX&yD,**RF*7$$\"39S2N'>.&**>F*$\"3CEo@KI,**RF*7$$\"3o?b()=L]**>F* $\"3CAU>AN,**RF*7$$\"3?3cXXM]**>F*$\"3Y[))QGS,**RF*7$$\"3mG&*GpN]**>F* $\"3ik:gBX,**RF*7$$\"3E9_5*o.&**>F*$\"3Eh`u-],**RF*7$$\"3#\\5S@#Q]**>F *$\"3=))GvMb,**RF*7$$\"3ErynTR]**>F*$\"3-W`y7g,**RF*7$$\"3s^*4$pS]**>F *$\"3-cq=Bl,**RF*7$$\"3S'ej\\=/&**>F*$\"3y&*oo&)p,**RF*7$$\"35dNS6V]** >F*$\"3;89K\"\\F*$\"3Q=T3nz,**RF*7$$\"3w6=taX] **>F*$\"3qgKRk%=!**RF*7$$\"3s!e9jn/&**>F*$\"3-wQg]*=!**RF*7$$\"3Ms/g.[ ]**>F*$\"3Ij8if%>!**RF*7$$\"3A9>>E\\]**>F*$\"3SKd')\\*>!**RF*7$$\"3)*p $f:00&**>F*$\"3oQ9@^/-**RF*7$$\"3o>())e<0&**>F*$\"3[zdS[4-**RF*7$$\"3k zW8!H0&**>F*$\"3KpdF09-**RF*7$$\"3W8I2@a]**>F*$\"3co.!*G>-**RF*7$$\"3u )*f=Qb]**>F*$\"3A![OsRA!**RF*7$$\"3!y'G0jc]**>F*$\"3g)[!e'*G-**RF*7$$ \"3A%RmDy0&**>F*$\"3I[k^uL-**RF*7$$\"3'******4\"f]**>F*$\"3]RR7))Q-**R F*7S7$$\"3'*******=f]**>F*$\"3YMg6?R-(*RF*7$$\"3'QJ_2j2&**>F*$\"3UJnVl 2.(*RF*7$$\"3_\\)Q,74&**>F*$\"39O%=:sOq*RF*7$$\"3!)y(4^z5&**>F*$\"3yR0 w>M/(*RF*7$$\"3y-N=\"[7&**>F*$\"3m2'3C;]q*RF*7$$\"3a7OCfT^**>F*$\"3-\" p:I(o0(*RF*7$$\"3-T,-:d^**>F*$\"331Dh%4jq*RF*7$$\"37Am$fK<&**>F*$\"31S 2sO&pq*RF*7$$\"3[^([>**=&**>F*$\"3c^P;*>wq*RF*7$$\"3uezh_1_**>F*$\"3!z uX-%G3(*RF*7$$\"3]g.#3OA&**>F*$\"3)yi?9n*3(*RF*7$$\"3slfSlQ_**>F*$\"3U 9!G$)o&4(*RF*7$$\"3C(*H@fb_**>F*$\"3M\\g%>Y-r*RF*7$$\"3;j\\(*fs_**>F*$ \"3)Q)HQj#4r*RF*7$$\"3ygU(*)*)G&**>F*$\"3y>K$y\"e6(*RF*7$$\"3)>kUtQI&* *>F*$\"3;;m!*p<7(*RF*7$$\"390m:d@`**>F*$\"3iYL]Z)Gr*RF*7$$\"3)*y6UcO`* *>F*$\"3Q\\5;V[8(*RF*7$$\"3;(HA/SN&**>F*$\"3Sc>a<=9(*RF*7$$\"3C'zeT%p` **>F*$\"3!p*e0\"*z9(*RF*7$$\"3%\\S#)yjQ&**>F*$\"3CBYQkZ:(*RF*7$$\"3/6y q]-a**>F*$\"31c1?97;(*RF*7$$\"3yD>_L>a**>F*$\"3a,E\"R%z;(*RF*7$$\"3-,1 ()yMa**>F*$\"3+YR*Q7ur*RF*7$$\"3I;ztX^a**>F*$\"3+lS%)*y!=(*RF*7$$\"3c# fWr(oa**>F*$\"3U#p)*Qr(=(*RF*7$$\"37%>RVQ[&**>F*$\"3#>[88u$>(*RF*7$$\" 3ka1:7+b**>F*$\"3I$o\"4^-?(*RF*7$$\"3'[F*$\"3Y8^Lwp?(*RF*7$$ \"3!\\IV!RLb**>F*$\"33I#zcb8s*RF*7$$\"3[&*G'3$\\b**>F*$\"3+>\"Q:#*>s*R F*7$$\"3K`rI)pc&**>F*$\"37/]u*)pA(*RF*7$$\"3+HWV'Ge&**>F*$\"3G!f[3MLs* RF*7$$\"3)[F*4#)*f&**>F*$\"3S_G,A,C(*RF*7$$\"3i]Oi=:c**>F*$\"3)*>0wmiC (*RF*7$$\"3cU1X)>j&**>F*$\"3#R;)f%)HD(*RF*7$$\"3)*Hp.zZc**>F*$\"3u1]c0 $fs*RF*7$$\"3UD]?Jkc**>F*$\"3g,?!G\"fE(*RF*7$$\"3oF8]Y!o&**>F*$\"3qN\" *esBF(*RF*7$$\"3SeocP(p&**>F*$\"37&>$RN\"zs*RF*7$$\"3E_REm8d**>F*$\"3m )o#y[cG(*RF*7$$\"3mo>%=.t&**>F*$\"3i#op'4BH(*RF*7$$\"3/e\"GOou&**>F*$ \"3][mS:*)H(*RF*7$$\"3U=\"[9?w&**>F*$\"3I'o(R&)\\I(*RF*7$$\"3yy5/Tzd** >F*$\"3%)z!)HU>J(*RF*7$$\"3OgP&p\\z&**>F*$\"3]`yjk\"=t*RF*7$$\"37*>\") e:\"e**>F*$\"3a#>b*)zCt*RF*7$$\"3SKkoVFe**>F*$\"3+h&[)[6L(*RF*7$$\"3'* ******\\We**>F*$\"3e-=oszL(*RF*7S7$$\"3'******zX%e**>F*$\"3W`^n/!Q$**R F*7$$\"336Txphe**>F*$\"3C+>N][M**RF*7$$\"3A7'z\"fwe**>F*$\"3;,Mu13N**R F*7$$\"3Cq=F*$\"3W.RL0vN**RF*7$$\"3OkqE?5f**>F*$\"3AzEL[UO**RF *7$$\"3OU&[$)p#f**>F*$\"3q/))Gf4P**RF*7$$\"3!>)[9aUf**>F*$\"3CE#47=x$* *RF*7$$\"39w=3lef**>F*$\"3kODlBOQ**RF*7$$\"35?_6Jvf**>F*$\"3Q$3UkG!R** RF*7$$\"3+ub!=>*f**>F*$\"3u/&py#pR**RF*7$$\"3oF(H+!4g**>F*$\"3%pp*RfPS **RF*7$$\"3C#\\MYS-'**>F*$\"3QV+iw(4%**RF*7$$\"3G#4j%)4/'**>F*$\"3-'R! f]lT**RF*7$$\"3M:nC*z0'**>F*$\"39&4\"Q_LU**RF*7$$\"3e$)oEQug**>F*$\"3c [A<2*H%**RF*7$$\"3SF*$\"3]J_bfeV**RF*7$$\"3?<2\\'p5'**>F*$\" 31\"4?v$HW**RF*7$$\"3U#Qud>7'**>F*$\"3-Z'*[L*[%**RF*7$$\"3S3xzRRh**>F* $\"3?8LB3fX**RF*7$$\"35lQb$[:'**>F*$\"3ud$o?3i%**RF*7$$\"3UT!*Hxrh**>F *$\"3#eQ\\d&)o%**RF*7$$\"3!\\)\\9!z='**>F*$\"3'H*3!fIv%**RF*7$$\"3[G0) HZ?'**>F*$\"3AsG'f.#[**RF*7$$\"3w\"))[$=?i**>F*$\"3abcE;#)[**RF*7$$\"3 uAuB&oB'**>F*$\"3U\"\\iD)[\\**RF*7$$\"3YYhm;ai**>F*$\"3mfs(p!=]**RF*7$ $\"3CS*zQ#pi**>F*$\"3;_bqMy]**RF*7$$\"31H@r^&G'**>F*$\"3oYB#[M9&**RF*7 $$\"3qj?UL-j**>F*$\"3-xbTq5_**RF*7$$\"3UVrky=j**>F*$\"3#R!>5]w_**RF*7$ $\"3-/q[qMj**>F*$\"3)***=H;S`**RF*7$$\"33oP&zBN'**>F*$\"3cAk'[3T&**RF* 7$$\"3qm75Eoj**>F*$\"3eZ.IOua**RF*7$$\"3%[q(y@&Q'**>F*$\"3MF*$\"3g!f%)GOg&**RF*7$$\"35H+=QF*$\"3%zkr53n&** RF*7$$\"3IVky=Lk**>F*$\"35hsO-Md**RF*7$$\"3Exb(4(\\k**>F*$\"31F*$\"3SU52qke**RF*7$$\"3y7&ztF['**>F*$\"3!H&oALKf **RF*7$$\"3GYt41*\\'**>F*$\"3mD^&pu*f**RF*7$$\"3#>d'pr:l**>F*$\"3/q&)= 3kg**RF*7$$\"3#\\z.NA`'**>F*$\"3)z6pU,8'**RF*7$$\"3u#3V8ua'**>F*$\"3Ar ed%3>'**RF*7$$\"3#\\>e4[c'**>F*$\"3w6\"Q=/E'**RF*7$$\"35*o!*o.e'**>F*$ \"3KC:]kAj**RF*7$$\"3__#Repf'**>F*$\"3yJR;**)Q'**RF*7$$\"3_/Zm$Gh'**>F *$\"3#Qd(Q\\_k**RF*7$$\"3-+++!*Hm**>F*$\"3#*Rddt?l**RF*7S7$$\"3-+++)*H m**>F*$\"3+[.d0@l(*RF*7$$\"396Tx4Zm**>F*$\"3;!)f^^*ew*RF*7$$\"3I7'z\"* >m'**>F*$\"3/P993\\m(*RF*7$$\"3Iq=F*$\"33Z]*pgrw*RF*7$$\"3WkqEg &p'**>F*$\"3-v'e-Nyw*RF*7$$\"3WU&[$Q7n**>F*$\"3?$Ry91&o(*RF*7$$\"3'>)[ 9%zs'**>F*$\"3q*>VOG\"p(*RF*7$$\"3Aw=30Wn**>F*$\"3mb&Rjs(p(*RF*7$$\"3; ?_6rgn**>F*$\"3I.3R*Q/x*RF*7$$\"31ub!=tx'**>F*$\"3K&3z5.6x*RF*7$$\"3uF (H+Wz'**>F*$\"3c0w(G'yr(*RF*7$$\"3I#\\MY%4o**>F*$\"3#eHM.)Qs(*RF*7$$\" 3O#4j%QEo**>F*$\"3u92da1t(*RF*7$$\"3S:nCRVo**>F*$\"3/t&GmXPx*RF*7$$\"3 m$)oEyfo**>F*$\"3i$=x;,Wx*RF*7$$\"3[F*$\"3CiRHk*\\x*RF*7$$\"3 G<2\\O#*o**>F*$\"3%y#o`Uqv(*RF*7$$\"3]#Qudt!p**>F*$\"3U\"*=uQIw(*RF*7$ $\"3[3xzzCp**>F*$\"3o3&fP,qx*RF*7$$\"3;lQbBSp**>F*$\"3OYq$y=wx*RF*7$$ \"3]T!*HF*$\"3GGTyhHy(*RF*7$$\"3'\\)\\9Itp**>F*$\"3?\")*)=7%*y(* RF*7$$\"3aG0)H,*p**>F*$\"39+`^Uhz(*RF*7$$\"3#=))[$e0q**>F*$\"3sH31BB!y *RF*7$$\"3!GUP_A-(**>F*$\"3g+&>'*)*3y*RF*7$$\"3_YhmcRq**>F*$\"3%>C1V\" f\"y*RF*7$$\"3IS*zQY0(**>F*$\"3G(GrA%>#y*RF*7$$\"37H@r\"42(**>F*$\"3s \"yVEXGy*RF*7$$\"3wj?Ut(3(**>F*$\"3)\\<,&y^$y*RF*7$$\"3]Vrk=/r**>F*$\" 3#Q$fWe<%y*RF*7$$\"33/q[5?r**>F*$\"33wf)[7[y*RF*7$$\"39oP&zx8(**>F*$\" 3eK\"QP>by*RF*7$$\"3wm75m`r**>F*$\"3QB:UX:'y*RF*7$$\"3#\\q(yhqr**>F*$ \"3k^[?F$oy*RF*7$$\"3cY;L)f=(**>F*$\"3;&[8DZuy*RF*7$$\"3=H+=y-s**>F*$ \"3=8W'4>\")y*RF*7$$\"3QVkye=s**>F*$\"3?3$3D^()y*RF*7$$\"3Kxb(4^B(**>F *$\"3!**[[.7%*y*RF*7$$\"3[[CHE^s**>F*$\"39\\`s!e+z*RF*7$$\"3'G^zt\"os* *>F*$\"3g&zYTM2z*RF*7$$\"3MYt4Y%G(**>F*$\"3s248eQ\"z*RF*7$$\"3+slp6,t* *>F*$\"3g%)fi>0#z*RF*7$$\"3+&z.NwJ(**>F*$\"3'y*f'f7Fz*RF*7$$\"3#G3V8GL (**>F*$\"3It6^'>Lz*RF*7$$\"3+&>e4-N(**>F*$\"3esm/a,%z*RF*7$$\"3=*o!*od O(**>F*$\"3S\"\\apPYz*RF*7$$\"3e_#ReBQ(**>F*$\"3U'[x=,`z*RF*7$$\"3g/Zm B)R(**>F*$\"31W0Ni$fz*RF*7$$\"35+++I:u**>F*$\"3aTn!o=mz*RF*7S7$$\"35++ +Q:u**>F*$\"3]0E!)=i'***RF*7$$\"3A6Tx\\Ku**>F*$\"3YBr,lI(***RF*7$$\"3Q 7'z\"RZu**>F*$\"3'e`w=-z***RF*7$$\"3Qq=<9ku**>F*$\"3m`K*4s&)***RF*7$$ \"3]kqE+\"[(**>F*$\"3IL<_kC****RF*7$$\"3]U&[$y(\\(**>F*$\"3iW]+w\"**** *RF*7$$\"3/#)[9M8v**>F*$\"3kNUT)R0++%F*7$$\"3Gw=3XHv**>F*$\"33QOOT=,+S F*7$$\"3C?_66Yv**>F*$\"3r&ewY]=++%F*7$$\"39ub!=Fc(**>F*$\"3GHdiY^-+SF* 7$$\"3#ysH+)zv**>F*$\"3axDpy>.+SF*7$$\"3;#\\MY[f(**>F*$\"3U4cQ'*z.+SF* 7$$\"3U#4j%y6w**>F*$\"3S'4))3xW++%F*7$$\"3[:nCzGw**>F*$\"3W8J@t:0+SF*7 $$\"3s$)oE=Xw**>F*$\"33#=>&G\"e++%F*7$$\"3cF*$\"3Lc(p83k++%F* 7$$\"3O<2\\wxw**>F*$\"3+G1*)f62+SF*7$$\"3c#QudFp(**>F*$\"3w)>Jj:x++%F* 7$$\"3K3xz>5x**>F*$\"3@mFiJT3+SF*7$$\"3ClQbjDx**>F*$\"3N)zUfI!4+SF*7$$ \"3MT!*HdUx**>F*$\"3zKf:!3(4+SF*7$$\"3/&)\\9qex**>F*$\"3RKT\"3`.,+%F*7 $$\"3SG0)Hbx(**>F*$\"3`!z/9E5,+%F*7$$\"3!>))[$)4z(**>F*$\"3GnI>Uk6+SF* 7$$\"3)GUP_w!y**>F*$\"3rsN,4J7+SF*7$$\"3gYhm'\\#y**>F*$\"3s'GsR.I,+%F* 7$$\"3QS*zQ+%y**>F*$\"3z&3u@1O,+%F*7$$\"3?H@rJcy**>F*$\"3pzA!GdU,+%F*7 $$\"3ij?U8ty**>F*$\"3UNQ#*)H\\,+%F*7$$\"3cVrke*)y**>F*$\"3kEq7ze:+SF*7 $$\"3;/q[]0z**>F*$\"3b:r\"eCi,+%F*7$$\"3+oP&zJ#z**>F*$\"3k/p%\\Jp,+%F* 7$$\"3%oE,h!Rz**>F*$\"3ci(zomv,+%F*7$$\"3)\\q(y,cz**>F*$\"3&)[%H*[C=+S F*7$$\"3SY;LQrz**>F*$\"3>U%zWf)=+SF*7$$\"3-H+==))z**>F*$\"3$4C%>8`>+SF *7$$\"3AVky)R+)**>F*$\"3zF*$\"35Eh3V#3-+% F*7$$\"3a[CHmO!)**>F*$\"3G>nr.Z@+SF*7$$\"3q7&ztN0)**>F*$\"3!3!QSn9A+SF *7$$\"3?Yt4')p!)**>F*$\"3#=vV;)zA+SF*7$$\"3%=d'p^'3)**>F*$\"3vg/SVYB+S F*7$$\"3%[z.NI5)**>F*$\"3!)R****\\7C+SF*7$$\"3)G3V8#=\")**>F*$\"3HQNy? tC+SF*7$$\"3%[>e4c8)**>F*$\"3)eH#fyUD+SF*7$$\"3C*o!*o6:)**>F*$\"3T@Xu, 0E+SF*7$$\"3W_#Rex;)**>F*$\"3l-\"Gp8n-+%F*7$$\"3W/Zmj$=)**>F*$\"3*ed]w [t-+%F*7$$\"3%*******p+#)**>F*$\"3>0[P7.G+SF*7S7$$\"3%******z2?)**>F*$ \"3#f#>PW.G)*RF*7$$\"316Tx*y@)**>F*$\"3!)G`&3>(G)*RF*7$$\"3A7'z\"zK#)* *>F*$\"3;(p[z9$H)*RF*7$$\"3Aq=F*$\"3;B&Gt%)*H)*RF*7$$\"3OkqES m#)**>F*$\"3]a=7\"f1$)*RF*7$$\"3OU&[$=$G)**>F*$\"3'*e(oGI8$)*RF*7$$\"3 !>)[9u)H)**>F*$\"3/MB_D&>$)*RF*7$$\"39w=3&[J)**>F*$\"33#yC(ofK)*RF*7$$ \"33?_6^J$)**>F*$\"3eI%*HKEL)*RF*7$$\"3)Rd0=\"[$)**>F*$\"3sN%4XFR$)*RF *7$$\"3mF(H+_O)**>F*$\"3c6Y%o5Y$)*RF*7$$\"3A#\\MY-Q)**>F*$\"3;))RxC@N) *RF*7$$\"3G#4j%=(R)**>F*$\"33SDa**)e$)*RF*7$$\"3K:nC>9%)**>F*$\"3w;Z8- dO)*RF*7$$\"3e$)oEeI%)**>F*$\"37U#)pdAP)*RF*7$$\"3SF*$\"3Y7E y5#y$)*RF*7$$\"3?<2\\;j%)**>F*$\"3w*[\"e*G&Q)*RF*7$$\"3U#Qud\"y%)**>F* $\"37ovD'G\"R)*RF*7$$\"3S3xzf&\\)**>F*$\"3+)3B=E)R)*RF*7$$\"33lQb.6&)* *>F*$\"3%>h&QOWS)*RF*7$$\"3UT!*H(z_)**>F*$\"37,['3@6%)*RF*7$$\"3)[)\\9 5W&)**>F*$\"3gXjxhwT)*RF*7$$\"3YG0)H4c)**>F*$\"3IW8j#RC%)*RF*7$$\"3u\" ))[$Qw&)**>F*$\"3'oOiOdI%)*RF*7$$\"3sAuB0$f)**>F*$\"3I2ZuSsV)*RF*7$$\" 3YYhmO5')**>F*$\"3)RRvf;W%)*RF*7$$\"3AS*zQai)**>F*$\"3)e%RT%>]%)*RF*7$ $\"3/H@rrT')**>F*$\"39SyH0nX)*RF*7$$\"3oj?U`e')**>F*$\"3AfNoJMY)*RF*7$ $\"3UVrk)\\n)**>F*$\"3Q#=X@,q%)*RF*7$$\"3+/q[!4p)**>F*$\"3i;`3zjZ)*RF* 7$$\"31oP&z&3()**>F*$\"3]SF\\[M[)*RF*7$$\"3om75YC()**>F*$\"3wj]n+)*[)* RF*7$$\"3%[q(yTT()**>F*$\"3+46*He'\\)*RF*7$$\"3[Y;Lyc()**>F*$\"3/jCyGF ])*RF*7$$\"35H+=et()**>F*$\"3[K6wZ%4&)*RF*7$$\"3IVkyQ*y)**>F*$\"3u!f,) pd^)*RF*7$$\"3Cxb(4f!))**>F*$\"3MC3;yB_)*RF*7$$\"3S[CH1A))**>F*$\"3W^^ /R)G&)*RF*7$$\"3y7&zt*Q))**>F*$\"3!)py*HgN&)*RF*7$$\"3EYt4Eb))**>F*$\" 3sfO\\<@a)*RF*7$$\"3#>d'p\">())**>F*$\"3;,?^z([&)*RF*7$$\"3#\\z.N%)))) **>F*$\"3UY4P'Qb&)*RF*7$$\"3u#3V8O!*)**>F*$\"3?mHRd9c)*RF*7$$\"3#\\>e4 5#*)**>F*$\"3)H)\\Z:%o&)*RF*7$$\"35*o!*ol$*)**>F*$\"3Y8;()QYd)*RF*7$$ \"3]_#ReJ&*)**>F*$\"3e$y:VF\"e)*RF*7$$\"3_/Zm.p*)**>F*$\"3UswGDwe)*RF* 7$$\"3-+++5')*)**>F*$\"3?K*z-X%f)*RF*7S7$$\"3-+++=')*)**>F*$\"3;5$yA[% f+SF*7$$\"3!RJ_(H.!***>F*$\"3O9M%*G8g+SF*7$$\"3c\\)Q\">=!***>F*$\"3#)z [>'G21+%F*7$$\"3')y(4T\\.***>F*$\"3F-Dv&)Rh+SF*7$$\"3#G]$=!=0***>F*$\" 344[sH2i+SF*7$$\"3e7OCeo!***>F*$\"3IP)\\;WF1+%F*7$$\"33T,-9%3***>F*$\" 3Gc&oWmL1+%F*7$$\"3=Am$\\-5***>F*$\"3'3+Uy5S1+%F*7$$\"3a^([4p6***>F*$ \"3e%\\$frnk+SF*7$$\"3!)ezh^L\"***>F*$\"3Ox(zRT`1+%F*7$$\"3cg.#)f]\"** *>F*$\"3hwi\\Y-m+SF*7$$\"3wlfSkl\"***>F*$\"3Xh`ekim+SF*7$$\"3G(*H@e#=* **>F*$\"3n3P`RIn+SF*7$$\"3Aj\\(*e*>***>F*$\"3F=kIU)z1+%F*7$$\"3%3Euzf@ ***>F*$\"3M@R/)R'o+SF*7$$\"3/UEM'3B***>F*$\"34ziG^Bp+SF*7$$\"3?0m:c[#* **>F*$\"36=IFI%*p+SF*7$$\"3-z6Ubj#***>F*$\"3HS#3rU02+%F*7$$\"3A(HA%*4G ***>F*$\"3I#))eGS72+%F*7$$\"3G'zeJkH***>F*$\"3Fq_ex&=2+%F*7$$\"3)\\S#) oLJ***>F*$\"3kVUC_`s+SF*7$$\"356yq\\H$***>F*$\"3I')pK.=t+SF*7$$\"3#e#> _KY$***>F*$\"3+61OM&Q2+%F*7$$\"33,1(yF*$\"3-pcb:Zu+SF*7$$\"3O;zt Wy$***>F*$\"3eU\\\"GQ^2+%F*7$$\"3i#fWhdR***>F*$\"3#HTH#3$e2+%F*7$$\"3= %>RL3T***>F*$\"3u\\z#oLk2+%F*7$$\"3qa1:6F%***>F*$\"36JY)y%3x+SF*7$$\"3 !\\F*$\"3%e&)[Wdx2+%F*7$$\"3'\\IV!Qg%***>F*$\"348^3bTy+SF*7$$ \"3a&*G')Hw%***>F*$\"3\\8U>A0z+SF*7$$\"3Q`rI(R\\***>F*$\"3VY#*y\"f(z+S F*7$$\"31HWV&)4&***>F*$\"3F*$\"3KyhjE2 \"3+%F*7$$\"3m]OiF*$\"3dG1fso\"3+%F*7$$\"3iU1X(*e&***>F*$\"3;0w u\"fB3+%F*7$$\"3-Ip.yu&***>F*$\"3lPe&R\"*H3+%F*7$$\"3YD]?I\"f***>F*$\" 3`U/\\Al$3+%F*7$$\"3uF8]X2'***>F*$\"39Fia$)H%3+%F*7$$\"3YeocOC'***>F*$ \"3.X%ywu\\3+%F*7$$\"3K_RElS'***>F*$\"3W9rMii&3+%F*7$$\"3so>%3tl***>F* $\"3X]AaCH'3+%F*7$$\"35e\"GEQn***>F*$\"3qDldJ&p3+%F*7$$\"3[=\"[/!*o*** >F*$\"3:c'fFgv3+%F*7$$\"3#)y5/S1(***>F*$\"3+Bj-hD)3+%F*7$$\"3UgP&f>s** *>F*$\"3:2\")e%y))3+%F*7$$\"3;*>\")[&Q(***>F*$\"3(RO3-U&*3+%F*7$$\"3YK koUa(***>F*$\"3x\"z[8x,4+%F*7$$\"3++++\\r(***>F*$\"3Yo@_'f34+%F*7S7$$ \"3++++dr(***>F*$\"3u-=_G'3*)*RF*7$$\"376Txo)y***>F*$\"3AtHava\"*)*RF* 7$$\"3G7'z\"e.)***>F*$\"3S]U5L9#*)*RF*7$$\"3Gq=F*$\"3Q)G5I8G*) *RF*7$$\"3UkqE>P)***>F*$\"3)*>LLx[$*)*RF*7$$\"3UU&[tR&)***>F*$\"3?2ug* eT*)*RF*7$$\"3%>)[9`p)***>F*$\"3?[(\\F\"y%*)*RF*7$$\"3?w=3k&))***>F*$ \"3=&GekDa*)*RF*7$$\"39?_6I-****>F*$\"3YJjb?4'*)*RF*7$$\"3/ub!3*=****> F*$\"3Cb!)Gjv'*)*RF*7$$\"3sF(H!*f$****>F*$\"3g$))fhRu*)*RF*7$$\"31#\\M O5&****>F*$\"3)H%>c9/)*)*RF*7$$\"3M#4juz'****>F*$\"3oDE')*=()*)*RF*7$$ \"3Q:nC)\\)****>F*$\"3s;\"*)H*R**)*RF*7$$\"3k$)oEP,++?F*$\"3K`v1\\0+** RF*7$$\"3YE]1!**RF*7$$\"3[<2\\&R.++#F*$\"3[*Ru>e8!** RF*7$$\"3E#QuZ*[++?F*$\"3w([@\"z&>!**RF*7$$\"3Y3xzQm++?F*$\"3-3\\Bbl-* *RF*7$$\"39lQb#=3++#F*$\"37:CGIF.**RF*7$$\"3ET!*Hw)4++#F*$\"3m1PH0&R!* *RF*7$$\"3s%)\\9*[6++#F*$\"3=R>rcf/**RF*7$$\"3`G0)><8++#F*$\"3U:c4)o_! **RF*7$$\"3\"=))[tr9++#F*$\"3UF@hp)e!**RF*7$$\"3yAuB%Q;++#F*$\"3IM\"=s `l!**RF*7$$\"3HYhm:\"=++#F*$\"3wiF*HYs!**RF*7$$\"31S*zGi>++#F*$\"3o<[! >\\y!**RF*7$$\"36H@r]7-+?F*$\"334,I.]3**RF*7$$\"3`j?UKH-+?F*$\"3)*\\T@ I<4**RF*7$$\"3DVrkxX-+?F*$\"3#Rj#>6$)4**RF*7$$\"32/q[ph-+?F*$\"3yeGjyY 5**RF*7$$\"38oP&p$z-+?F*$\"33Ybf[<6**RF*7$$\"3`m75D&H++#F*$\"3C)zw75= \"**RF*7$$\"3!\\q(y?7.+?F*$\"3Mdb7%)[7**RF*7$$\"3KY;LdF.+?F*$\"3OG'*RI 58**RF*7$$\"3$*G+=PW.+?F*$\"3mNg!*\\x8**RF*7$$\"38Vky2R;**RF*7$$\"35Yt40E/+?F*$\"3uyX?A/<* *RF*7$$\"3wrlpqU/+?F*$\"3g\"=YZ3x\"**RF*7$$\"3w%z.D#f/+?F*$\"3ybS7#p$= **RF*7$$\"3z#3V.WZ++#F*$\"3me*z\"\\++#F*$\"31Z9DA n>**RF*7$$\"3;*o!*et]++#F*$\"3)p)o8YH?**RF*7$$\"37_#R[R_++#F*$\"3wFA5# e4#**RF*7$$\"38/Zm#)R0+?F*$\"39XHdLf@**RF*7$$\"3')******)ob++#F*$\"3ck 75fFA**RF*7S7$$\"33+++(pb++#F*$\"3^c@5\"zA7+%F*7$$\"3)46u(3u0+?F*$\"3Y 6ARQ'H7+%F*7$$\"387'z\")*)e++#F*$\"3iWu='fN7+%F*7$$\"3Oq=E,SF*7$$\"30w=3/r1+?F*$\"3_ i/$3Uo7+%F*7$$\"3A?_6q(o++#F*$\"385->&3v7+%F*7$$\"3!Rd03Vq++#F*$\"3f%z #=G3+?F*$\"3/&Hw'[xK,SF*7$$\"3L#QuZV$3+?F*$\"3#4*)eguL8+ %F*7$$\"3`3xzy^3+?F*$\"3#=EYCsS8+%F*7$$\"3@lQbAn3+?F*$\"3&HEOx*oM,SF*7 $$\"3LT!*H;%)3+?F*$\"3X2O,tON,SF*7$$\"3z%)\\9H+4+?F*$\"3l'=&oC,O,SF*7$ $\"3gG0)>r\"4+?F*$\"3m,KLcoO,SF*7$$\"3)=))[tD$4+?F*$\"3\"3Y#4QIP,SF*7$ $\"3'GUPU#\\4+?F*$\"3o-.'fqz8+%F*7$$\"3OYhmbm4+?F*$\"3\"Q!p+KmQ,SF*7$$ \"38S*zG;)4+?F*$\"3d6d:hER,SF*7$$\"3=H@r!z*4+?F*$\"3K.n!G<*R,SF*7$$\"3 gj?Us95+?F*$\"392\\)***eS,SF*7$$\"3KVrkfU,SF*7$$\"3gm 75l!3,+#F*$\"3BLJ3sAV,SF*7$$\"3(\\q(yg(4,+#F*$\"3$3D)>b!R9+%F*7$$\"3RY ;L(H6,+#F*$\"3o\"o8X,SF*7$$\"3?Vkyd X6+?F*$\"3Xf#pUCe9+%F*7$$\"3:xb(*4i6+?F*$\"3:nqS`[Y,SF*7$$\"3`[CHDy6+? F*$\"3C%f_]Jr9+%F*7$$\"3p7&zj^>,+#F*$\"3/;A!)z!y9+%F*7$$\"3=Yt4X67+?F* $\"3,?b1&f%[,SF*7$$\"3$=d'p5G7+?F*$\"3[a(oyD\"\\,SF*7$$\"3$[z.DYC,+#F* $\"3)[41b'y\\,SF*7$$\"3(G3V.)f7+?F*$\"3\\yLCPR],SF*7$$\"3$[>e*>x7+?F*$ \"3`n^9'*3^,SF*7$$\"3B*o!*eFH,+#F*$\"3$G,v-7<:+%F*7$$\"3>_#R[$48+?F*$ \"3hT4]cP_,SF*7$$\"3?/ZmAD8+?F*$\"3pt5A3,`,SF*7$$\"3$*******GU8+?F*$\" 30Cu,Mp`,SF*7S7$$\"3:+++PU8+?F*$\"3?t&>g'p`**RF*7$$\"306Tx[f8+?F*$\"3_ 8&yN\"Qa**RF*7$$\"3?7'z\"Qu8+?F*$\"3w,xgr(\\&**RF*7$$\"3Vq=<8\"R,+#F*$ \"37c*R?Zc&**RF*7$$\"3ckqE*zS,+#F*$\"3+#p#*o@j&**RF*7$$\"3yU&[tZU,+#F* $\"3'f'RpH*p&**RF*7$$\"34#)[9LS9+?F*$\"35v]K`hd**RF*7$$\"37w=3Wc9+?F*$ \"3!GqRvf#e**RF*7$$\"3H?_65t9+?F*$\"3I^6;i#*e**RF*7$$\"3(Rd03(*[,+#F*$ \"3I(f9a!ff**RF*7$$\"35G(H!z1:+?F*$\"3g#3B)QFg**RF*7$$\"3V#\\MO=_,+#F* $\"3wGypd(3'**RF*7$$\"3[#4ju(Q:+?F*$\"3eV1`Lbh**RF*7$$\"3I:nCyb:+?F*$ \"3a_9>PBi**RF*7$$\"3y$)oEk**RF*7$$\"3S#QuZ(>;+?F*$ \"3WdLLDzk**RF*7$$\"3g3xz=P;+?F*$\"3)*yY*>!\\l**RF*7$$\"3HlQbi_;+?F*$ \"3Etr_x5m**RF*7$$\"3ST!*Hcp;+?F*$\"3gr02`ym**RF*7$$\"3'[)\\9p&o,+#F*$ \"3e'\\&*\\Iu'**RF*7$$\"3nG0)>Dq,+#F*$\"3E^y!p.\"o**RF*7$$\"3&>))[tzr, +#F*$\"3oc)4*=so**RF*7$$\"3$HUPUYt,+#F*$\"3aL&Rq)Qp**RF*7$$\"3VYhm&>v, +#F*$\"3M2\"eL\"3q**RF*7$$\"3?S*zGqw,+#F*$\"3%)oOuUoq**RF*7$$\"3DH@rI$ y,+#F*$\"3/g.laLr**RF*7$$\"3nj?U7+=+?F*$\"3yEF4#3?(**RF*7$$\"3SVrkd;=+ ?F*$\"3gu!)ejms**RF*7$$\"3@/q[\\K=+?F*$\"3)=RG:.L(**RF*7$$\"3FoP&p,&=+ ?F*$\"37Yj/-,u**RF*7$$\"3nm750m=+?F*$\"3sIlAbku**RF*7$$\"3/0xy+$)=+?F* $\"3!o+3'QKv**RF*7$$\"3YY;LP)*=+?F*$\"3O)zk`Qf(**RF*7$$\"33H+=<:>+?F*$ \"31Z*)R0hw**RF*7$$\"3FVky(4$>+?F*$\"3AGDVGCx**RF*7$$\"3Axb(*\\Z>+?F*$ \"3mh)Hy.z(**RF*7$$\"3g[CHlj>+?F*$\"3oA\"H(*\\&y**RF*7$$\"3w7&zj0)>+?F *$\"3)3QWZE#z**RF*7$$\"3DYt4&o*>+?F*$\"3'G_j-y)z**RF*7$$\"3!>d'p]8?+?F *$\"3s!RGLW0)**RF*7$$\"3!\\z.D+.-+#F*$\"3Y'>D707)**RF*7$$\"3%H3V._/-+# F*$\"3o-4?B\"=)**RF*7$$\"3!\\>e*fi?+?F*$\"3Y]fP#3D)**RF*7$$\"3I*o!*e\" y?+?F*$\"3/--v18$)**RF*7$$\"3F_#R[Z4-+#F*$\"3/-+# F*$\"3Itz<0u(=+%F*7$$\"3&Ga[t,@-+#F*$\"3'*RGC=T)=+%F*7$$\"3<#)[9tDA+?F *$\"30M$=@M!*=+%F*7$$\"3>w=3%=C-+#F*$\"3M2ge'y'*=+%F*7$$\"3O?_6]eA+?F* $\"3rc\"p9X.>+%F*7$$\"3/ub!3^F-+#F*$\"3SjM)\\45>+%F*7$$\"3#H-+#F *$\"3Ew-mGp\">+%F*7$$\"3]#\\MOsI-+#F*$\"3\"eOrx%H#>+%F*7$$\"3b#4juTK-+ #F*$\"3\"yCqQsH>+%F*7$$\"3Q:nC=TB+?F*$\"3^:#)zFl$>+%F*7$$\"3&Q)oEddB+? F*$\"3[C!\\Y3V>+%F*7$$\"3o+%F*7$$\"3q<2\\:!R-+ #F*$\"3Cw74>h&>+%F*7$$\"3Z#QuZ^S-+#F*$\"3N()[%p6i>+%F*7$$\"3n3xzeAC+?F *$\"3Ug,)Q4p>+%F*7$$\"3OlQb-QC+?F*$\"3%p9b'p_(>+%F*7$$\"3ZT!*H'\\X-+#F *$\"37*fka/#)>+%F*7$$\"3%\\)\\94rC+?F*$\"3!*pGk(\\))>+%F*7$$\"3uG0)>z[ -+#F*$\"38l&>)H_*>+%F*7$$\"3-#))[tL]-+#F*$\"3\"oJk?T,?+%F*7$$\"3+BuB/? D+?F*$\"3zFeX!33?+%F*7$$\"3]YhmNPD+?F*$\"3Duj/2],-SF*7$$\"3GS*zGCb-+#F *$\"3Z*oom.@?+%F*7$$\"3KH@rqoD+?F*$\"39!3J)[v--SF*7$$\"3uj?U_&e-+#F*$ \"3o5w`wU.-SF*7$$\"3ZVrk(>g-+#F*$\"3+!R\"He3/-SF*7$$\"3G/q[*yh-+#F*$\" 3g_<[Es/-SF*7$$\"3MoP&pbj-+#F*$\"3jTtF(Ha?+%F*7$$\"3um75X^E+?F*$\"3d\" *pq]11-SF*7$$\"360xySoE+?F*$\"3/F[NMu1-SF*7$$\"3`Y;Lx$o-+#F*$\"3UyHN\" et?+%F*7$$\"3:H+=d+F+?F*$\"3#y*4l,.3-SF*7$$\"3NVkyP;F+?F*$\"3QgG$\\i'3 -SF*7$$\"3Ixb(**Gt-+#F*$\"3W?(*eMK4-SF*7$$\"3n[CH0\\F+?F*$\"3\\9Fu'p*4 -SF*7$$\"3$G^zjfw-+#F*$\"344O-ik5-SF*7$$\"3KYt4D#y-+#F*$\"3***e)zxH6-S F*7$$\"3(>d'p!*)z-+#F*$\"3L!4D6k>@+%F*7$$\"3(\\z.Da\"G+?F*$\"3Sh8G\\i7 -SF*7$$\"3,$3V.1$G+?F*$\"3D!\\&\\@B8-SF*7$$\"3(\\>e**z%G+?F*$\"3y'zV4G R@+%F*7$$\"3P*o!*eN'G+?F*$\"3u`Cc0b9-SF*7$$\"3M_#R[,)G+?F*$\"3jd&4B9_@ +%F*7$$\"3N/Zm-'*G+?F*$\"3!4_GX\\e@+%F*7$$\"32+++48H+?F*$\"3-M4'3Kl@+% F*7S7$$\"3&)*****pJ\"H+?F*$\"3g$flGNl,+%F*7$$\"3v5TxGIH+?F*$\"3_0B'4?s ,+%F*7$$\"3!>hz\"=XH+?F*$\"3p0%f%f\"y,+%F*7$$\"38q=<$>'H+?F*$\"3evyTg[ =+SF*7$$\"3EkqEzyH+?F*$\"3&[J+eg\">+SF*7$$\"3[U&[tb*H+?F*$\"3Au(G\">$) >+SF*7$$\"3z\")[986I+?F*$\"3F`'[Ka/-+%F*7$$\"3Ew=3CFI+?F*$\"3![Ppz)4@+ SF*7$$\"3**>_6!R/.+#F*$\"3^@U6`w@+SF*7$$\"36ub!301.+#F*$\"3_\"R*)oHC-+ %F*7$$\"3zF(H!fxI+?F*$\"33JX$38J-+%F*7$$\"38#\\MOE4.+#F*$\"3-k>=]rB+SF *7$$\"3=#4ju&4J+?F*$\"3I7paERC+SF*7$$\"3X:nCeEJ+?F*$\"3_S?uI2D+SF*7$$ \"3[$)oE(H9.+#F*$\"3x0.&yGd-+%F*7$$\"3I.+#F*$\"3HzM*3Kw-+%F*7$$\"3I 3xz)z?.+#F*$\"3A,F5)H$G+SF*7$$\"3)\\'QbUBK+?F*$\"3x\"=?TZ*G+SF*7$$\"3a T!*HOSK+?F*$\"3p)o&>]iH+SF*7$$\"3,&)\\9\\cK+?F*$\"3D0ti-FI+SF*7$$\"3PG 0)>LF.+#F*$\"3OQ$o]V4.+%F*7$$\"3l\"))[t()G.+#F*$\"3?PebL[[.+%F*7$$\"3aVrkP(Q.+#F*$\"3Vn0xy!QX.+#F*$\"3J4(QCk\"Q+SF*7$$\"3gY;L&)e7&QV+SF*7$$\"30&z.D3g.+#F*$\"3G*eu'f/W+SF* 7$$\"33$3V.gh.+#F*$\"3&)Rr7KlW+SF*7$$\"3/&>e*RLO+?F*$\"380([=\\`/+%F*7 $$\"3W*o!*e*[O+?F*$\"3PoPNm/+%F*7$ $\"3U/ZmU\"o.+#F*$\"37Ry=1FZ+SF*7$$\"39+++\\)p.+#F*$\"3<$G)yK&z/+%F*7S 7$$\"3#******p&)p.+#F*$\"3'3?%zk&zC+%F*7$$\"3f8Bvo:P+?F*$\"3M#fsIT'[-S F*7$$\"3Z\\)Q\"eIP+?F*$\"3kP+?F*$\"3)HTD%=e]-SF*7$$\"3r7OC(4y.+#F*$\"3)p(>$>`7D+%F *7$$\"3?T,-`'z.+#F*$\"3$3)p@c(=D+%F*7$$\"3IAm$RE\"Q+?F*$\"3\\y'36?DD+% F*7$$\"3W^([*HHQ+?F*$\"3v^.Vm=`-SF*7$$\"3[ezh!f%Q+?F*$\"35y(*H@(\\*Q+?F*$\"34o+cS\"eD+%F*7$$\"37j\\(z>\"R+?F*$\"3!)3d$\\%\\c- SF*7$$\"3ugU(p$GR+?F*$\"3zJz@-:d-SF*7$$\"3sTEMDVR+?F*$\"3+VI'oXxD+%F*7 $$\"350m:&4'R+?F*$\"3`$z'ze@)3S+?F*$\"34!pT$*o.E+%F*7$$ \"3n/C)ed-/+#F*$\"3))fpfl/h-SF*7$$\"3B6yq)=//+#F*$\"3&ov*>=ph-SF*7$$\" 3&f#>_reS+?F*$\"3U'R>3lBE+%F*7$$\"3?,1(oT2/+#F*$\"3h64ZL)HE+%F*7$$\"3[ ;zt$34/+#F*$\"3=r6I-lj-SF*7$$\"3u#fW^\"3T+?F*$\"3VauMHMk-SF*7$$\"3I%>R BK7/+#F*$\"3v)[m$f%\\E+%F*7$$\"3ga1:]RT+?F*$\"3dXt&>(fl-SF*7$$\"3.v\"R =j:/+#F*$\"3ZEl5+Fm-SF*7$$\"3&[IVqFHWVCAU+? F*$\"3R!>6g2*o-SF*7$$\"30v#*4?RU+?F*$\"3]^`5gep-SF*7$$\"3z]OicaU+?F*$ \"3.Uz]2?q-SF*7$$\"3_U1XOrU+?F*$\"3'z6[#G(3F+%F*7$$\"3:Ip.<(G/+#F*$\"3 S=g%>0:F+%F*7$$\"3PD]?p.V+?F*$\"3fax.i;s-SF*7$$\"3'yK,X)>V+?F*$\"3U@fh C\"GF+%F*7$$\"3eeocvOV+?F*$\"3(>$>M!*[t-SF*7$$\"3A_RE/`V+?F*$\"3s6ca19 u-SF*7$$\"3io>%)ppV+?F*$\"3#y^5.2[F+%F*7$$\"3Ae\"G;iQ/+#F*$\"3bl:!*yYv -SF*7$$\"3g=\"[%R,W+?F*$\"3g3_^^2w-SF*7$$\"3&*y5/z=W+?F*$\"34.9U6xw-SF *7$$\"3JgP&\\VV/+#F*$\"3++'\\k$Rx-SF*7$$\"3H*>\")Q4X/+#F*$\"3#3OLOd!y- SF*7$$\"3OKko\"oY/+#F*$\"37f-FEpy-SF*7$$\"38+++)Q[/+#F*$\"3>*z^Iv$z-SF *7S7$$\"3\"******fR[/+#F*$\"3Ks*e]y$z+SF*7$$\"3WV.#3_\\/+#F*$\"3@T.N&G )z+SF*7$$\"3850_*\\]/+#F*$\"3-e=.,A!3+%F*7$$\"3\\17<+;X+?F*$\"3!*Rvi/m !3+%F*7$$\"3NIx63FX+?F*$\"3*4c:u.63+%F*7$$\"3fx$)z5QX+?F*$\"3L]x8\\a\" 3+%F*7$$\"3C!\\CJ$[X+?F*$\"3mJ6PR&>3+%F*7$$\"3)\\'Ro\"*eX+?F*$\"3-(3tX xB3+%F*7$$\"3x(oZk)pX+?F*$\"3_ktia\"G3+%F*7$$\"3n*[+x2e/+#F*$\"3zPrj?D $3+%F*7$$\"31G\")=+#f/+#F*$\"3jCth6q$3+%F*7$$\"3'3&e())=g/+#F*$\"3H/sF n4%3+%F*7$$\"3\"p0/>Ih/+#F*$\"3\"Qn:/UX3+%F*7$$\"31kC]>CY+?F*$\"3u^;%= *)\\3+%F*7$$\"3t9c^'\\j/+#F*$\"3#*y9*3?a3+%F*7$$\"3qNnauWY+?F*$\"32ZN# R6e3+%F*7$$\"3NN._PcY+?F*$\"3FT'**owi3+%F*7$$\"3w@:rAmY+?F*$\"3bTGe3n' 3+%F*7$$\"3I!3B(oxY+?F*$\"38**)*p$Hr3+%F*7$$\"3m$\\PJyo/+#F*$\"3;/wI_` (3+%F*7$$\"3L\")36'*)p/+#F*$\"30!)eC0)z3+%F*7$$\"3cTZ#f&4Z+?F*$\"3n[%) \\XS)3+%F*7$$\"3zA$H<1s/+#F*$\"3-n&f(p%))3+%F*7$$\"3SBJ?xIZ+?F*$\"3ELX hKD*3+%F*7$$\"3L.)GD 4(44+%F*7$$\"3GhGRv%y/+#F*$\"3bY,^IT\"4+%F*7$$\"3m3L[c&z/+#F*$\"3;bw!f X=4+%F*7$$\"3-0R\\-1[+?F*$\"3xwV&4kA4+%F*7$$\"3%HL5Rw\"[+?F*$\"3/(zPxG F4+%F*7$$\"3C\\Y\\2G[+?F*$\"3\"Rn\"3j9$4+%F*7$$\"3>bOu@R[+?F*$\"3Hu[:@ f$4+%F*7$$\"3&[')=9$\\[+?F*$\"3@SR$3'*R4+%F*7$$\"3!*e0ENg[+?F*$\"3t4DF xV%4+%F*7$$\"3\")o&*)Q2([+?F*$\"3X[#*zK&[4+%F*7$$\"3n^hbf\")[+?F*$\"3r eV_wG&4+%F*7$$\"3*zn$*4A*[+?F*$\"3\">*=JBr&4+%F*7$$\"3H*R?AL!\\+?F*$\" 3J$G2$p:'4+%F*7$$\"3I/QY-9\\+?F*$\"3*zfJ8&e'4+%F*7$$\"3;kA%p\\#\\+?F*$ \"3E%HA.Bq4+%F*7$$\"3V!*yN#e$\\+?F*$\"3%e5bIdu4+%F*7$$\"3'Q7U(zX\\+?F* $\"3];wdj&y4+%F*7$$\"3s4q&Gs&\\+?F*$\"3_*=pr8$)4+%F*7$$\"3D\"fs_u'\\+? F*$\"3=Ii%yA()4+%F*7$$\"3p[3QNy\\+?F*$\"36hMO*e\"*4+%F*7$$\"3PXNvy))\\ +?F*$\"3)>>%*Qw&*4+%F*7$$\"3<+++++]+?F*$\"3f+++]-+,SF*-%'COLOURG6&%$RG BG$\"*++++\"!\")$\"\"!F]cxF\\cx-%*THICKNESSG6#\"\"\"-%+AXESLABELSG6$Q \"x6\"Q!Ffcx-%%VIEWG6$;$\"&&**>!\"%$\"&0+#F^dx%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "If the option " } {TEXT 262 12 "discont=true" }{TEXT -1 25 " is omitted from Maple's " } {TEXT 0 4 "plot" }{TEXT -1 174 " procedure then all sample points obta ined for the plot are joined with line segments. This leads to approxi mately vertical line segments being drawn at the discontinuities. " }} {PARA 0 "" 0 "" {TEXT -1 17 "Using the option " }{TEXT 262 12 "discont =true" }{TEXT -1 114 " causes Maple to search for discontinuities and \+ then break up the graph into the appropriate disconnected pieces. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "d := .0005;\nplot(f(x),x=2-d..2+d);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dG$\"\"&!\"%" }}{PARA 13 "" 1 "" {GLPLOT2D 415 275 275 {PLOTDATA 2 "6%-%'CURVESG6$7[w7$$\"31+++++]**>!#<$\"3#*******\\-+**RF* 7$$\"3#4FWYs-&**>F*$\"3;j)fe96!**RF*7$$\"3cT&)G\\a]**>F*$\"3KsXtT?-**R F*7$$\"3M4'\\/81&**>F*$\"3?qbqlZ-(*RF*7$$\"37x1h6o]**>F*$\"3+'\\x'*[Fq *RF*7$$\"3mWF*$\"3g[.l8-.(*RF*7$$\"3W7G$R<3&**>F*$\"3EITiPH. (*RF*7$$\"3wZ\\DO&4&**>F*$\"3AwWd&QQq*RF*7$$\"3I$3x&)*3^**>F*$\"3?N&GN $Q/(*RF*7$$\"3/Dc'yM;&**>F*$\"3E()=QDc1(*RF*7$$\"3ymT:(z@&**>F*$\"37HY HF*$\"39Q(*[R`7(*RF*7$$\"3Ie9ui2a**>F*$\"31$p k=Ejr*RF*7$$\"3k;z_\"4i&**>F*$\"3Q'\\vyb[s*RF*7$$\"3!om6m#Gd**>F*$\"3' 4xB*)[\"H(*RF*7$$\"3u;aphNe**>F*$\"3sGD??WL(*RF*7$$\"3Qx#\\%HUe**>F*$ \"3GgAm!4Pt*RF*7$$\"3/QJ?(*[e**>F*$\"3!Q)G7h(R$**RF*7$$\"3o)*p&\\c&e** >F*$\"3G*R%eJCM**RF*7$$\"3Mf3rKie**>F*$\"3%p!o/-^M**RF*7$$\"3k!e=#ove* *>F*$\"3a'HuHW]$**RF*7$$\"3;-js.*)e**>F*$\"3%[N0Ryb$**RF*7$$\"3)\\uTZd \"f**>F*$\"3Mr\"ydYm$**RF*7$$\"3e(=ddC%f**>F*$\"3)fDlw9x$**RF*7$$\"3+t !)y(e*f**>F*$\"3YJA[6&)R**RF*7$$\"3We*=)H\\g**>F*$\"3u\"Gc`()>%**RF*7$ $\"3+v=JN[h**>F*$\"3%pL*f*[f%**RF*7$$\"3!=z/3uC'**>F*$\"3uI'QS5*\\**RF *7$$\"3We*ot*\\j**>F*$\"3GaFqA,a**RF*7$$\"33DJ$RDX'**>F*$\"3=ssdT6e**R F*7$$\"3g\"z4wb]'**>F*$\"3?]\"\\DN-'**RF*7$$\"3MekGhel**>F*$\"3?(GxNcB '**RF*7$$\"3q\"zCJ^e'**>F*$\"3%GX7\"pTj**RF*7$$\"31DJ'\\;h'**>F*$\"3M# ohYxW'**RF*7$$\"3I3F#z#=m**>F*$\"3;(=^gUZ'**RF*7$$\"3w\"H#)3\\i'**>F*$ \"3Ir:Wx+l**RF*7$$\"3?v=%Q:j'**>F*$\"3SNG$)GFl(*RF*7$$\"3We9!o\"Qm**>F *$\"3!o(\\A!Qbw*RF*7$$\"37D1sU^m**>F*$\"3O)*=,$ogw*RF*7$$\"3!=zR'okm** >F*$\"3yNB!e)fm(*RF*7$$\"3M3_(>/x'**>F*$\"3I>F?s#3x*RF*7$$\"35D1J:wo** >F*$\"3g%pE)e0v(*RF*7$$\"3[L3En$4(**>F*$\"37B2^`v$y*RF*7$$\"37Dc#o%*=( **>F*$\"3K(Q$Hme(y*RF*7$$\"3w;/RE&G(**>F*$\"3!yef#zT\"z*RF*7$$\"3_P4b= Rt**>F*$\"3o(3.]uNz*RF*7$$\"3Ge9r5$R(**>F*$\"3QQZ!3Jdz*RF*7$$\"3OB:t%) *R(**>F*$\"3GLS`1+'z*RF*7$$\"3Y)e^(e1u**>F*$\"3APUE-F'z*RF*7$$\"3c`;xK 8u**>F*$\"3M\\`*zRlz*RF*7$$\"3k=F*$\"3%)pts$4o***RF*7$$\"3%)[= $[NV(**>F*$\"3aPT>&[t***RF*7$$\"3-z>(GqW(**>F*$\"3mRXmw)y***RF*7$$\"3S RA&*)RZ(**>F*$\"3uYihf'*)***RF*7$$\"3++D.&4](**>F*$\"3K$\\#eU/++SF*7$$ \"3-+]jB4w**>F*$\"3[3vp^P/+SF*7$$\"3/+vB_F*$\"3$4/Z51(3+SF*7$$\"3 /+v'Hi#z**>F*$\"3mL_([`q,+%F*7$$\"3/Y'y#*4-)**>F*$\"3-!GziV3-+%F*7$$\" 3%=z*ev:\")**>F*$\"3$e#H'yLY-+%F*7$$\"3)=U_ER9)**>F*$\"37v(f]gd-+%F*7$ $\"3;_]r4s\")**>F*$\"31(\\sA()o-+%F*7$$\"3InjC='=)**>F*$\"3r4[)e]u-+%F *7$$\"3W#oxn-?)**>F*$\"3u!4,&R,G+SF*7$$\"3-SL/J2#)**>F*$\"3=>2JcHG)*RF *7$$\"3e(**3`V@)**>F*$\"3[R87tdG)*RF*7$$\"3;bYdR@#)**>F*$\"3W^H$**e)G) *RF*7$$\"3s7.%Q%G#)**>F*$\"3%fbXnS\"H)*RF*7$$\"31tb'zZG)**>F*$\"3s+@GT RJ)*RF*7$$\"3SL347T$)**>F*$\"3/K@)eZO$)*RF*7$$\"3G$3xxlV)**>F*$\"31C7b bYP)*RF*7$$\"3SLLY.K&)**>F*$\"3)pb-a$GT)*RF*7$$\"3=D\"o7Tv)**>F*$\"37` [fg;])*RF*7$$\"3?HK5S_))**>F*$\"31^Fet4a)*RF*7$$\"3YL$Q*o]*)**>F*$\"3c hQw'G!e)*RF*7$$\"3Kq?\"pT'*)**>F*$\"3QPxPyce)*RF*7$$\"3S2e)[w(*)**>F*$ \"3/[_**p5f)*RF*7$$\"3%fns)Q%)*)**>F*$\"3Im`!ew$f)*RF*7$$\"3YW&fG6**)* *>F*$\"3$HR;;Y'f+SF*7$$\"3+8k%oy**)**>F*$\"3RG$Gu:*f+SF*7$$\"3a\"GL3Y+ ***>F*$\"3dr6C`=g+SF*7$$\"3[b2ycJ!***>F*$\"3CJ;]OEh+SF*7$$\"3SH#GF&e!* **>F*$\"3$oiw(>Mi+SF*7$$\"3ExJiW7\"***>F*$\"3\"eAqj)\\k+SF*7$$\"37D\"= lj;***>F*$\"3Fq>-`lm+SF*7$$\"3G]iB0p#***>F*$\"3#e%yGEwq+SF*7$$\"3?vV&R F*$\"368Yw*p[2+%F*7$$\"3WzWG))y%***>F*$\"3^[N&eb\"z+SF*7$$\"3Y$e 9Ege***>F*$\"39x?<7W$3+%F*7$$\"3=FW;AN'***>F*$\"3eT$))**3a3+%F*7$$\"3! 4F9F*$\"334I&ywt3+%F*7$$\"3)G>*[,4(***>F*$\"3`%\\.og$)3+%F*7$$ \"3i9TEhL(***>F*$\"3`!3mdW$*3+%F*7$$\"3gv::\"fu***>F*$\"3c7>Dl$)*3+%F* 7$$\"3QO!R5#e(***>F*$\"3&zwSZG.4+%F*7$$\"3)pw#)fVw***>F*$\"3uIj[Wd!4+% F*7$$\"3O(\\E40x***>F*$\"3P\\EB/#34+%F*7$$\"3uF-(emx***>F*$\"3]C(zRm5* )*RF*7$$\"3MeR\"3Gy***>F*$\"3ecvsBJ\"*)*RF*7$$\"3Y(=7O*)))***>F*$\"3%> 5sXdb*)*RF*7$$\"3!oT5k]*****>F*$\"356>kD!)**)*RF*7$$\"3j\"zRQb@++#F*$ \"3eMP#e@'3**RF*7$$\"3VLe,]6.+?F*$\"3=dO.,Y7**RF*7$$\"3Av=>Y2/+?F*$\"3 O`xU')H;**RF*7$$\"3z&Qe#Gf/+?F*$\"3M[H9:P=**RF*7$$\"3\"f*[K560+?F*$\"3 #)[=\"RW/#**RF*7$$\"3AB:%eS_++#F*$\"3+lC6E'4#**RF*7$$\"3(4:e8q`++#F*$ \"3GSkJ3[@**RF*7$$\"32lk6\\V0+?F*$\"3!po>%*R<#**RF*7$$\"3syZ(o*\\0+?F* $\"33rP_!**>#**RF*7$$\"3Q#4LYkb++#F*$\"3g%pG;eA#**RF*7$$\"3.19R#Hc++#F *$\"3mdWts^A,SF*7$$\"3ahYU$))e++#F*$\"33/f;PbB,SF*7$$\"3g;zXu91+?F*$\" 3Jv2h,fC,SF*7$$\"3he9i\"=s++#F*$\"3%3-'pJ()G,SF*7$$\"3=+]y))G3+?F*$\"3 S^0,i:L,SF*7$$\"3'*\\i_QQ5+?F*$\"3)*Ru)[O:9+%F*7$$\"3c(=-N(R6+?F*$\"3! *4()*p!fX,SF*7$$\"3;D\"y%3T7+?F*$\"3[:aJ\\k\\,SF*7$$\"3BG)o<#p7+?F*$\" 3f[W=.x],SF*7$$\"3tJ&f]tH,+#F*$\"3g7$pq&*=:+%F*7$$\"3/$)[qT68+?F*$\"3T zw,%eC:+%F*7$$\"3![B]$[D8+?F*$\"3$[+q4@I:+%F*7$$\"3*3\"Hn^K8+?F*$\"38_ wWCI`,SF*7$$\"3b'e&*\\&R8+?F*$\"34(GEz$e`,SF*7$$\"3?i#=$eY8+?F*$\"3C6f S^'Q&**RF*7$$\"3'y$4kh`8+?F*$\"31Dl)[YT&**RF*7$$\"3*RMA#))49+?F*$\"35b qwsRc**RF*7$$\"36]P![hY,+#F*$\"3i,4r!['e**RF*7$$\"3-U5FEn:+?F*$\"3;#HZ 'Hpi**RF*7$$\"3[L$Qx$o;+?F*$\"3Ol\")yytm**RF*7$$\"3N+v.I%)=+?F*$\"3k!) elbPv**RF*7$$\"3(fk`H@)>+?F*$\"3AfH5\"*Gz**RF*7$$\"3f\"zpe*z?+?F*$\"3A [9uE?$)**RF*7$$\"3r5PyK$4-+#F*$\"33d]&\\PP)**RF*7$$\"3GIwpp1@+?F*$\"3m UAM&=+%F*7$$\"3B2LN+%F*7$$\"3Lzp%*\\%R-+#F*$\"3s%>Cr&y&>+%F*7$$\"3Se9S8&\\- +#F*$\"3KrF'e6)*>+%F*7$$\"3>D1#=bq-+#F*$\"3^)y![!G#3-SF*7$$\"3&RUB@+%F*7$$\"3fL3s?6H+?F*$\"3:4YjnX;-SF*7$$\"3zn*\\Oz\"H+? F*$\"3\"[SV(fs;+SF*7$$\"3a,\"zlY#H+?F*$\"3J0J&=&*p,+%F*7$$\"3tN#3&RJH+ ?F*$\"3`8P'Rks,+%F*7$$\"3#*ptV7QH+?F*$\"3oF_2O`<+SF*7$$\"3JQcHe^H+?F*$ \"30s4I?2=+SF*7$$\"3q1R:/lH+?F*$\"3HR.`/h=+SF*7$$\"3.V/(e>*H+?F*$\"3kS ***H(o>+SF*7$$\"3Nzpe()=I+?F*$\"3jKS[Tw?+SF*7$$\"3,_+-rsI+?F*$\"3g(o& \\y\"H-+%F*7$$\"37DJXaEJ+?F*$\"3(oIlbr]-+%F*7$$\"3s&*[ACIK+?F*$\"3zKUC ,AH+SF*7$$\"3wmm*RRL.+#F*$\"3,C#QroL.+%F*7$$\"3%=/'e)*RM+?F*$\"38%>zE6 w.+%F*7$$\"3\"pTvJga.+#F*$\"3Wz]WQ&=/+%F*7$$\"3OFWch)f.+#F*$\"3ka!eddR /+%F*7$$\"3!yV`*>^O+?F*$\"3RJj781Y+SF*7$$\"3>!p]XVm.+#F*$\"36SqZseY+SF *7$$\"3-Vz9\\xO+?F*$\"3R27$=8r/+%F*7$$\"3mplW1%o.+#F*$\"3A)e4:wt/+%F*7 $$\"3&e>XP1p.+#F*$\"3ZJ))=\"Rw/+%F*7$$\"30AQ/@(p.+#F*$\"3.Q*o3-z/+%F*7 $$\"3C[CMy.P+?F*$\"3X4*\\0l\"[-SF*7$$\"3Y`p`2IP+?F*$\"37PCGp@\\-SF*7$$ \"3oe9tOcP+?F*$\"3G*yG!)o-D+%F*7$$\"3]H#e0I&Q+?F*$\"3GQ%*o]8a-SF*7$$\" 3J+]Qk\\R+?F*$\"3xmo`8+e-SF*7$$\"3h$3dg6!\"%$\"&0+#Fbhn%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 44.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 41 "More examples of discontinuou s functions " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "The function f given by " } {XPPEDIT 18 0 "f(x) = x+abs(x)/x;" "6#/-%\"fG6#%\"xG,&F'\"\"\"*&-%$abs G6#F'F)F'!\"\"F)" }{TEXT -1 24 " has a discontinuity at " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 15 ". The value of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 14 " are close to " }{XPPEDIT 18 0 "-1" "6#,$\"\"\"!\"\"" }{TEXT -1 6 " when " }{TEXT 267 1 "x" } {TEXT -1 53 " is close to 0 and negative, but jump to values close" } {XPPEDIT 18 0 " ``+1" "6#,&%!G\"\"\"F%F%" }{TEXT -1 4 " as " }{TEXT 268 1 "x" }{TEXT -1 59 " increases through 0 to take positive values c lose to zero." }}{PARA 0 "" 0 "" {TEXT -1 22 "For example, f(-0.001)" }{XPPEDIT 18 0 "`` = -1.001;" "6#/%!G,$-%&FloatG6$\"%,5!\"$!\"\"" } {TEXT -1 16 ", while f(0.001)" }{XPPEDIT 18 0 "``=1.001" "6#/%!G-%&Flo atG6$\"%,5!\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "f := x -> x+abs(x)/x:\n'f(x) '=f(x);\nplot(f(x),x=-3..3,discont=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&F'\"\"\"*&-%$absGF&F)F'!\"\"F)" }} {PARA 13 "" 1 "" {GLPLOT2D 334 366 366 {PLOTDATA 2 "6%-%'CURVESG6%7S7$ $!\"$\"\"!$!\"%F*7$$!3*GyIw`3Y$H!#<$!3*GyIw`3Y$RF07$$!3Ew&pex6x(GF0$!3 Ew&pex6x(QF07$$!3;\\Di;as8GF0$!3;\\Di;as8QF07$$!3?q8D9\\J\\FF0$!3?q8D9 \\J\\PF07$$!3yyXvV0@&o#F0$!3yyXvV0@&o$F07$$!3sWMP'exdi#F0$!3sWMP'exdi$ F07$$!3[Bl\\,#QUc#F0$!3[Bl\\,#QUc$F07$$!3p6Qi\"3%f+DF0$!3p6Qi\"3%f+NF0 7$$!3*=p]#pS:PCF0$!3*=p]#pS:PMF07$$!3m.iv=#)*=P#F0$!3m.iv=#)*=P$F07$$! 3ie67I3U9BF0$!3ie67I3U9LF07$$!3Sq0+/\\r\\AF0$!3Sq0+/\\r\\KF07$$!3S808* GVZ=#F0$!3S808*GVZ=$F07$$!3%ytb#*4J@7#F0$!3%ytb#*4J@7$F07$$!3Q`W\\KKFl ?F0$!3Q`W\\KKFlIF07$$!3ksY]HPm(*>F0$!3'Gn/&HPm(*HF07$$!3y?#>@h*QS>F0$! 3y?#>@h*QSHF07$$!3ynu(y>mP(=F0$!3+ou(y>mP(GF07$$!3t8/P(=$z9=F0$!3t8/P( =$z9GF07$$!3=>[7[/4][7[/4]FF07$$!3qV!)\\R\"y%)o\"F0$!3qV!)\\R \"y%)o#F07$$!3t:;+f@>C;F0$!3t:;+f@>CEF07$$!3%[o%*4cd^c\"F0$!3i%o%*4cd^ c#F07$$!3lQqur2[,:F0$!3lQqur2[,DF07$$!30BVv$[Q`V\"F0$!30BVv$[Q`V#F07$$ !3=x%>wUhxP\"F0$!3=x%>wUhxP#F07$$!3pY)='Hmd:8F0$!3#p%)='Hmd:BF07$$!3)H L(\\[OL^7F0$!3)HL(\\[OL^AF07$$!3IJI([U%[)=\"F0$!3IJI([U%[)=#F07$$!30^' p$pXnF6F0$!3$3lp$pXnF@F07$$!31*oHEfb,1\"F0$!3G*oHEfb,1#F07$$!3Ko-,!*y' [***!#=$!3%o-,!*y'[**>F07$$!3q-eI;*)4Z$*Ffu$!3F!eI;*)4Z$>F07$$!3Ua(H([ R7g()Ffu$!3WvH([R7g(=F07$$!3\"p=j(o_S=\")Ffu$!3p=j(o_S=\"=F07$$!3!p!3A ,!)f9vFfu$!3q!3A,!)f9v\"F07$$!3G5J[FaW$)oFfu$!3-6$[FaW$)o\"F07$$!3uWsY A%yjE'Ffu$!3[CnCUyjE;F07$$!37p#4]Xm.i&Ffu$!3#p#4]Xm.i:F07$$!3MRO+],=)* \\Ffu$!3%RO+],=)*\\\"F07$$!3#>>w()y/>O%Ffu$!3>>w()y/>O9F07$$!3;.#)y3\" )*3t$Ffu$!3K?)y3\")*3t8F07$$!3]iyp.&o5:$Ffu$!3D'yp.&o5:8F07$$!3Okp_U$= l[#Ffu$!3W'p_U$=l[7F07$$!3%)Hd@cn8#*=Ffu$!3)Hd@cn8#*=\"F07$$!3G&>LP,-% e7Ffu$!3`>LP,-%e7\"F07$$!3TrKYC+P=l!#>$!3sKYC+P=l5F07$$!3'************ ***f!#E$!30+++1+++5F07S7$$\"3'***************fFhz$\"30+++1+++5F07$$\"3 z1<#p$o9RlFbz$\"3;<#p$o9Rl5F07$$\"3XNUI,B)GA\"Ffu$\"3cB/8I#)GA6F07$$\" 353Xx$*eui=Ffu$\"3)3Xx$*eui=\"F07$$\"3^)H'[<4&o]#Ffu$\"3')H'[<4&o]7F07 $$\"3O5UXAY*y9$Ffu$\"3/@aCi%*y98F07$$\"39^bE'>CAu$Ffu$\"37bli>CAu8F07$ $\"3YjZ.X!=wN%Ffu$\"3NwM]/=wN9F07$$\"3O\")=wV#fS*\\Ffu$\"39)=wV#fS*\\ \"F07$$\"3i\"3$\\n$f%GcFfu$\"3;3$\\n$f%Gc\"F07$$\"3/lzVsy,\"G'Ffu$\"3S 'zVsy,\"G;F07$$\"3_;%)ye5&G](Ffu $\"3))H%**>5&G]F07$$\"3UF`\\wiL- 5F0$\"3kF`\\wiL-?F07$$\"3Ez2)QR5'f5F0$\"3Ez2)QR5'f?F07$$\"3FKD73QBE6F0 $\"3\\KD73QBE@F07$$\"3K'eH'=o?&=\"F0$\"3K'eH'=o?&=#F07$$\"3)3=vyb4*\\7 F0$\"3)3=vyb4*\\AF07$$\"3Nc>]m=_68F0$\"3Nc>]m=_6BF07$$\"3K%Q)*p%y!eP\" F0$\"3K%Q)*p%y!eP#F07$$\"3@:`+XC%[V\"F0$\"3*\\J0]WU[V#F07$$\"3ShHDM#>& )\\\"F0$\"3ShHDM#>&)\\#F07$$\"3+xcCA:mk:F0$\"3+xcCA:mkDF07$$\"3)G_!Qy& QAi\"F0$\"3)G_!Qy&QAi#F07$$\"3O`6QwLU%o\"F0$\"3e`6QwLU%o#F07$$\"33nE]d jm[F0$\"3@6.P8W%)RHF07$$\"3At*)*p@8 0+#F0$\"3At*)*p@80+$F07$$\"3y>%pV6!Hl?F0$\"3y>%pV6!HlIF07$$\"3hCq76w)R 7#F0$\"3hCq76w)R7$F07$$\"3O\"oB\"z%f\")=#F0$\"3O\"oB\"z%f\")=$F07$$\"3 O>z(e?S&[AF0$\"3O>z(e?S&[KF07$$\"3.*o^KYb;J#F0$\"3.*o^KYb;J$F07$$\"3ev Kvj@OtBF0$\"3evKvj@OtLF07$$\"39t!*\\gL'zV#F0$\"39t!*\\gL'zV$F07$$\"37O '**4*>=+DF0$\"37O'**4*>=+NF07$$\"3'3QAr_4Qc#F0$\"3'3QAr_4Qc$F07$$\"3uz 67&>5pi#F0$\"3uz67&>5pi$F07$$\"3!Q@Ic:$*[o#F0$\"3!Q@Ic:$*[o$F07$$\"3i. tur\"[8v#F0$\"3i.tur\"[8v$F07$$\"32F%y.L'y5GF0$\"32F%y.L'y5QF07$$\"3_! oEY!)fT(GF0$\"3_!oEY!)fT(QF07$$\"3Mn`v0j\"[$HF0$\"3Mn`v0j\"[$RF07$$\" \"$F*$\"\"%F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$F*F*Fhjl-%+AXESLABELSG6$ Q\"x6\"Q!F][m-%%VIEWG6$;F(F]jl%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 9 "The term " }{XPPEDIT 18 0 "abs(x)/x" "6#*&-%$absG6#%\"xG \"\"\"F'!\"\"" }{TEXT -1 22 " has the value 1 when " }{TEXT 269 1 "x" }{TEXT -1 17 " is positive and " }{XPPEDIT 18 0 "-1" "6#,$\"\"\"!\"\" " }{TEXT -1 6 " when " }{TEXT 270 1 "x" }{TEXT -1 14 " is negative. " }{XPPEDIT 18 0 "abs(x)/x" "6#*&-%$absG6#%\"xG\"\"\"F'!\"\"" }{TEXT -1 21 " is not defined when " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" } {TEXT -1 42 " since it gives the meaningless exression " }{XPPEDIT 18 0 "0/0" "6#*&\"\"!\"\"\"F$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 14 "Indeed, since " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "abs(x) = PIECEWISE([x, 0 <= x],[-x, x < 0]);" "6#/-%$ab sG6#%\"xG-%*PIECEWISEG6$7$F'1\"\"!F'7$,$F'!\"\"2F'F-" }{TEXT -1 2 ", \+ " }}{PARA 0 "" 0 "" {TEXT -1 16 "it follows that " }}{PARA 257 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "abs(x)/x = PIECEWISE([1, 0 < x],[unde fined, x = 0],[-1, x < 0]);" "6#/*&-%$absG6#%\"xG\"\"\"F(!\"\"-%*PIECE WISEG6%7$F)2\"\"!F(7$%*undefinedG/F(F07$,$F)F*2F(F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "The mid dle line, \"" }{TEXT 273 9 "undefined" }{TEXT -1 5 " " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 13 "\", in this " }{TEXT 261 17 "piecewise formula" }{TEXT -1 45 " is included for emphasis, but ca n be omitted" }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }{XPPEDIT 18 0 "f( x)" "6#-%\"fG6#%\"xG" }{TEXT -1 54 " can be described by the following piecewise formula: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)=PIECEWISE([x+1,x>0],[undefined, x = 0],[x-1,x<0])" "6#/-%\"fG6 #%\"xG-%*PIECEWISEG6%7$,&F'\"\"\"F-F-2\"\"!F'7$%*undefinedG/F'F/7$,&F' F-F-!\"\"2F'F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 54 "Again, the middle line is only included for emphasis. " }}{PARA 0 "" 0 "" {TEXT -1 13 "The graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 53 " therefore consists of the part of the infinite line " } {XPPEDIT 18 0 "y=x+1" "6#/%\"yG,&%\"xG\"\"\"F'F'" }{TEXT -1 32 " which lies to the right of the " }{TEXT 271 1 "y" }{TEXT -1 41 " axis toget her with the part of the line " }{XPPEDIT 18 0 "y=x-1" "6#/%\"yG,&%\"x G\"\"\"F'!\"\"" }{TEXT -1 31 " which lies to the left of the " }{TEXT 272 1 "y" }{TEXT -1 62 " axis. The graph consists of these two \"half- lines\" or \"rays\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "plots[display]([plot(x+1,x=0..3),plot(x-1 ,x=-3..0)]);" }}{PARA 13 "" 1 "" {GLPLOT2D 307 325 325 {PLOTDATA 2 "6& -%'CURVESG6$7S7$$\"\"!F)$\"\"\"F)7$$\"3s******\\i9Rl!#>$\"31++]i9Rl5!# <7$$\"3/++vVA)GA\"!#=$\"34+]PC#)GA6F27$$\"3+++]Peui=F6$\"3%****\\Peui= \"F27$$\"3A++]i3&o]#F6$\"33++D'3&o]7F27$$\"3%)***\\(oX*y9$F6$\"3/+](oX *y98F27$$\"3z***\\P9CAu$F6$\"3#***\\P9CAu8F27$$\"3!)***\\P*zhdVF6$\"3. +]P*zhdV\"F27$$\"31++v$>fS*\\F6$\"3++]P>fS*\\\"F27$$\"3$)***\\(=$f%GcF 6$\"35+](=$f%Gc\"F27$$\"3Q+++Dy,\"G'F6$\"3$*****\\#y,\"G;F27$$\"33++]7 F27$$\"3,++]siL-5F2$\"3,++]siL-?F27$$ \"3-+++!R5'f5F2$\"3-+++!R5'f?F27$$\"3)***\\P/QBE6F2$\"3)***\\P/QBE@F27 $$\"3!******\\\"o?&=\"F2$\"3!******\\\"o?&=#F27$$\"31+]Pa&4*\\7F2$\"3% )**\\Pa&4*\\AF27$$\"33+]7j=_68F2$\"3&)**\\7j=_6BF27$$\"33++vVy!eP\"F2$ \"33++vVy!eP#F27$$\"34+](=WU[V\"F2$\"3K+](=WU[V#F27$$\"3)****\\7B>&)\\ \"F2$\"3)****\\7B>&)\\#F27$$\"3)***\\P>:mk:F2$\"3)***\\P>:mkDF27$$\"3' ***\\iv&QAi\"F2$\"3'***\\iv&QAi#F27$$\"31++vtLU%o\"F2$\"31++vtLU%o#F27 $$\"3!******\\Nm'[F2$\" 3=++D6W%)RHF27$$\"3z*****\\@80+#F2$\"3z*****\\@80+$F27$$\"31++]7,Hl?F2 $\"31++]7,HlIF27$$\"3()**\\P4w)R7#F2$\"3()**\\P4w)R7$F27$$\"3;++]x%f\" )=#F2$\"3;++]x%f\")=$F27$$\"3!)**\\P/-a[AF2$\"3!)**\\P/-a[KF27$$\"3/+] (=Yb;J#F2$\"3/+](=Yb;J$F27$$\"3')****\\i@OtBF2$\"3')****\\i@OtLF27$$\" 3')**\\PfL'zV#F2$\"3')**\\PfL'zV$F27$$\"3>+++!*>=+DF2$\"3>+++!*>=+NF27 $$\"3-++DE&4Qc#F2$\"3-++DE&4Qc$F27$$\"3=+]P%>5pi#F2$\"3=+]P%>5pi$F27$$ \"39+++bJ*[o#F2$\"39+++bJ*[o$F27$$\"33++Dr\"[8v#F2$\"33++Dr\"[8v$F27$$ \"3++++Ijy5GF2$\"3++++Ijy5QF27$$\"31+]P/)fT(GF2$\"31+]P/)fT(QF27$$\"31 +]i0j\"[$HF2$\"31+]i0j\"[$RF27$$\"\"$F)$\"\"%F)-%'COLOURG6&%$RGBG$\"#5 !\"\"F(F(-F$6$7S7$$!\"$F)$!\"%F)7$$!3&*****\\P&3Y$HF2$!3&*****\\P&3Y$R F27$$!3!***\\ivF2$!3)*****\\FPm(*HF27$$!3********4'*QS>F2$!3**** ****4'*QSHF27$$!3-+]i&>mP(=F2$!3-+]i&>mP(GF27$$!34+++&=$z9=F2$!34+++&= $z9GF27$$!3%***\\iX/4]C;F2$!3\"****\\i:#>CEF27$$!3!***\\7ev:l:F2$!3o **\\7ev:lDF27$$!3.++vo2[,:F2$!3.++vo2[,DF27$$!3-+]i![Q`V\"F2$!3-+]i![Q `V#F27$$!3/+]PC9wx8F2$!3/+]PC9wxBF27$$!3%****\\iiwbJ\"F2$!3%****\\iiwb J#F27$$!35+++XOL^7F2$!3))*****\\kL8D#F27$$!33++D@W[)=\"F2$!33++D@W[)=# F27$$!3-+]ilXnF6F2$!3-+]ilXnF@F27$$!3/++v)eb,1\"F2$!3#)***\\()eb,1#F27 $$!35-++]y'[***F6$!3?+++&y'[**>F27$$!3[*****\\())4Z$*F6$!3&*****\\())4 Z$>F27$$!3Q,+D1R7g()F6$!39+]i!R7g(=F27$$!3G)****\\A0%=\")F6$!3$)****\\ A0%=\"=F27$$!3/-+Dczf9vF6$!3?+]i&zf9v\"F27$$!3g***\\7QXM)oF6$!3'***\\7 QXM)o\"F27$$!3G,++v$yjE'F6$!38++]PyjE;F27$$!3R,+D1kO?cF6$!39+]iSm.i:F2 7$$!37)******4!=)*\\F6$!3#)******4!=)*\\\"F27$$!3u****\\PZ!>O%F6$!3(** **\\PZ!>O9F27$$!3E)**\\i0)*3t$F6$!3#)**\\i0)*3t8F27$$!3e)*****\\%o5:$F 6$!3')*****\\%o5:8F27$$!39****\\(G=l[#F6$!3#****\\(G=l[7F27$$!3+++++n8 #*=F6$!3++++qO@*=\"F27$$!3G***\\i&>Se7F6$!3$***\\i&>Se7\"F27$$!3V$*** \\P%p$=lF/$!3%***\\P%p$=l5F27$F($F_[lF)Fiz-%+AXESLABELSG6%Q\"x6\"Q!Fij l-%%FONTG6#%(DEFAULTG-%%VIEWG6$;Fd[lFezF^[m" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "To emphasise that neit her of the end points" }{XPPEDIT 18 0 "``(0,-1)" "6#-%!G6$\"\"!,$\"\" \"!\"\"" }{TEXT -1 4 " nor" }{XPPEDIT 18 0 "``(0,1)" "6#-%!G6$\"\"!\" \"\"" }{TEXT -1 90 " of the two \"half-lines\" belong to the graph the end points can be drawn as open circles. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "f := x -> x+abs(x )/x:\n'f(x)'=f(x);\np1 := plot(f(x),x=-3..3,discont=true):\np2 := plot ([[0,-1],[0,1]],style=point,color=red,symbol=circle):\nplots[display]( [p1,p2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&F'\"\"\"* &-%$absGF&F)F'!\"\"F)" }}{PARA 13 "" 1 "" {GLPLOT2D 315 318 318 {PLOTDATA 2 "6&-%'CURVESG6%7S7$$!\"$\"\"!$!\"%F*7$$!3*GyIw`3Y$H!#<$!3* GyIw`3Y$RF07$$!3Ew&pex6x(GF0$!3Ew&pex6x(QF07$$!3;\\Di;as8GF0$!3;\\Di;a s8QF07$$!3?q8D9\\J\\FF0$!3?q8D9\\J\\PF07$$!3yyXvV0@&o#F0$!3yyXvV0@&o$F 07$$!3sWMP'exdi#F0$!3sWMP'exdi$F07$$!3[Bl\\,#QUc#F0$!3[Bl\\,#QUc$F07$$ !3p6Qi\"3%f+DF0$!3p6Qi\"3%f+NF07$$!3*=p]#pS:PCF0$!3*=p]#pS:PMF07$$!3m. iv=#)*=P#F0$!3m.iv=#)*=P$F07$$!3ie67I3U9BF0$!3ie67I3U9LF07$$!3Sq0+/\\r \\AF0$!3Sq0+/\\r\\KF07$$!3S808*GVZ=#F0$!3S808*GVZ=$F07$$!3%ytb#*4J@7#F 0$!3%ytb#*4J@7$F07$$!3Q`W\\KKFl?F0$!3Q`W\\KKFlIF07$$!3ksY]HPm(*>F0$!3' Gn/&HPm(*HF07$$!3y?#>@h*QS>F0$!3y?#>@h*QSHF07$$!3ynu(y>mP(=F0$!3+ou(y> mP(GF07$$!3t8/P(=$z9=F0$!3t8/P(=$z9GF07$$!3=>[7[/4][7[/4]FF07$ $!3qV!)\\R\"y%)o\"F0$!3qV!)\\R\"y%)o#F07$$!3t:;+f@>C;F0$!3t:;+f@>CEF07 $$!3%[o%*4cd^c\"F0$!3i%o%*4cd^c#F07$$!3lQqur2[,:F0$!3lQqur2[,DF07$$!30 BVv$[Q`V\"F0$!30BVv$[Q`V#F07$$!3=x%>wUhxP\"F0$!3=x%>wUhxP#F07$$!3pY)=' Hmd:8F0$!3#p%)='Hmd:BF07$$!3)HL(\\[OL^7F0$!3)HL(\\[OL^AF07$$!3IJI([U%[ )=\"F0$!3IJI([U%[)=#F07$$!30^'p$pXnF6F0$!3$3lp$pXnF@F07$$!31*oHEfb,1\" F0$!3G*oHEfb,1#F07$$!3Ko-,!*y'[***!#=$!3%o-,!*y'[**>F07$$!3q-eI;*)4Z$* Ffu$!3F!eI;*)4Z$>F07$$!3Ua(H([R7g()Ffu$!3WvH([R7g(=F07$$!3\"p=j(o_S=\" )Ffu$!3p=j(o_S=\"=F07$$!3!p!3A,!)f9vFfu$!3q!3A,!)f9v\"F07$$!3G5J[FaW$) oFfu$!3-6$[FaW$)o\"F07$$!3uWsYA%yjE'Ffu$!3[CnCUyjE;F07$$!37p#4]Xm.i&Ff u$!3#p#4]Xm.i:F07$$!3MRO+],=)*\\Ffu$!3%RO+],=)*\\\"F07$$!3#>>w()y/>O%F fu$!3>>w()y/>O9F07$$!3;.#)y3\")*3t$Ffu$!3K?)y3\")*3t8F07$$!3]iyp.&o5:$ Ffu$!3D'yp.&o5:8F07$$!3Okp_U$=l[#Ffu$!3W'p_U$=l[7F07$$!3%)Hd@cn8#*=Ffu $!3)Hd@cn8#*=\"F07$$!3G&>LP,-%e7Ffu$!3`>LP,-%e7\"F07$$!3TrKYC+P=l!#>$! 3sKYC+P=l5F07$$!3'***************f!#E$!30+++1+++5F07S7$$\"3'********** *****fFhz$\"30+++1+++5F07$$\"3z1<#p$o9RlFbz$\"3;<#p$o9Rl5F07$$\"3XNUI, B)GA\"Ffu$\"3cB/8I#)GA6F07$$\"353Xx$*eui=Ffu$\"3)3Xx$*eui=\"F07$$\"3^) H'[<4&o]#Ffu$\"3')H'[<4&o]7F07$$\"3O5UXAY*y9$Ffu$\"3/@aCi%*y98F07$$\"3 9^bE'>CAu$Ffu$\"37bli>CAu8F07$$\"3YjZ.X!=wN%Ffu$\"3NwM]/=wN9F07$$\"3O \")=wV#fS*\\Ffu$\"39)=wV#fS*\\\"F07$$\"3i\"3$\\n$f%GcFfu$\"3;3$\\n$f%G c\"F07$$\"3/lzVsy,\"G'Ffu$\"3S'zVsy,\"G;F07$$\"3_;%)ye5&G](Ffu$\"3))H%**>5&G]F07$$\"3UF`\\wiL-5F0$\"3kF`\\wiL-?F07$$\"3Ez2)QR5'f5F0$ \"3Ez2)QR5'f?F07$$\"3FKD73QBE6F0$\"3\\KD73QBE@F07$$\"3K'eH'=o?&=\"F0$ \"3K'eH'=o?&=#F07$$\"3)3=vyb4*\\7F0$\"3)3=vyb4*\\AF07$$\"3Nc>]m=_68F0$ \"3Nc>]m=_6BF07$$\"3K%Q)*p%y!eP\"F0$\"3K%Q)*p%y!eP#F07$$\"3@:`+XC%[V\" F0$\"3*\\J0]WU[V#F07$$\"3ShHDM#>&)\\\"F0$\"3ShHDM#>&)\\#F07$$\"3+xcCA: mk:F0$\"3+xcCA:mkDF07$$\"3)G_!Qy&QAi\"F0$\"3)G_!Qy&QAi#F07$$\"3O`6QwLU %o\"F0$\"3e`6QwLU%o#F07$$\"33nE]djm[F0$\"3@6.P8W%)RHF07$$\"3At*)*p@80+#F0$\"3At*)*p@80+$F07$$\"3y>%pV6!H l?F0$\"3y>%pV6!HlIF07$$\"3hCq76w)R7#F0$\"3hCq76w)R7$F07$$\"3O\"oB\"z%f \")=#F0$\"3O\"oB\"z%f\")=$F07$$\"3O>z(e?S&[AF0$\"3O>z(e?S&[KF07$$\"3.* o^KYb;J#F0$\"3.*o^KYb;J$F07$$\"3evKvj@OtBF0$\"3evKvj@OtLF07$$\"39t!*\\ gL'zV#F0$\"39t!*\\gL'zV$F07$$\"37O'**4*>=+DF0$\"37O'**4*>=+NF07$$\"3'3 QAr_4Qc#F0$\"3'3QAr_4Qc$F07$$\"3uz67&>5pi#F0$\"3uz67&>5pi$F07$$\"3!Q@I c:$*[o#F0$\"3!Q@Ic:$*[o$F07$$\"3i.tur\"[8v#F0$\"3i.tur\"[8v$F07$$\"32F %y.L'y5GF0$\"32F%y.L'y5QF07$$\"3_!oEY!)fT(GF0$\"3_!oEY!)fT(QF07$$\"3Mn `v0j\"[$HF0$\"3Mn`v0j\"[$RF07$$\"\"$F*$\"\"%F*-%'COLOURG6&%$RGBG$\"*++ ++\"!\")$F*F*Fhjl-F$6&7$7$Fhjl$!\"\"F*7$Fhjl$\"\"\"F*Fajl-%'SYMBOLG6#% 'CIRCLEG-%&STYLEG6#%&POINTG-%+AXESLABELSG6%Q\"x6\"Q!F^\\m-%%FONTG6#%(D EFAULTG-%%VIEWG6$;F(F]jlFc\\m" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 201 "f := x -> x+abs(x )/x:\n'f(x)'=f(x);\np1 := plot(f(x),x=-3..3,discont=true):\np2 := plot ([[[0,-1],[0,1]]$2],style=point,color=red,symbol=[circle$2],symbolsize =[12,16]):\nplots[display]([p1,p2],thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&F'\"\"\"*&-%$absGF&F)F'!\"\"F)" }} {PARA 13 "" 1 "" {GLPLOT2D 319 342 342 {PLOTDATA 2 "6(-%'CURVESG6%7S7$ $!\"$\"\"!$!\"%F*7$$!3*GyIw`3Y$H!#<$!3*GyIw`3Y$RF07$$!3Ew&pex6x(GF0$!3 Ew&pex6x(QF07$$!3;\\Di;as8GF0$!3;\\Di;as8QF07$$!3?q8D9\\J\\FF0$!3?q8D9 \\J\\PF07$$!3yyXvV0@&o#F0$!3yyXvV0@&o$F07$$!3sWMP'exdi#F0$!3sWMP'exdi$ F07$$!3[Bl\\,#QUc#F0$!3[Bl\\,#QUc$F07$$!3p6Qi\"3%f+DF0$!3p6Qi\"3%f+NF0 7$$!3*=p]#pS:PCF0$!3*=p]#pS:PMF07$$!3m.iv=#)*=P#F0$!3m.iv=#)*=P$F07$$! 3ie67I3U9BF0$!3ie67I3U9LF07$$!3Sq0+/\\r\\AF0$!3Sq0+/\\r\\KF07$$!3S808* GVZ=#F0$!3S808*GVZ=$F07$$!3%ytb#*4J@7#F0$!3%ytb#*4J@7$F07$$!3Q`W\\KKFl ?F0$!3Q`W\\KKFlIF07$$!3ksY]HPm(*>F0$!3'Gn/&HPm(*HF07$$!3y?#>@h*QS>F0$! 3y?#>@h*QSHF07$$!3ynu(y>mP(=F0$!3+ou(y>mP(GF07$$!3t8/P(=$z9=F0$!3t8/P( =$z9GF07$$!3=>[7[/4][7[/4]FF07$$!3qV!)\\R\"y%)o\"F0$!3qV!)\\R \"y%)o#F07$$!3t:;+f@>C;F0$!3t:;+f@>CEF07$$!3%[o%*4cd^c\"F0$!3i%o%*4cd^ c#F07$$!3lQqur2[,:F0$!3lQqur2[,DF07$$!30BVv$[Q`V\"F0$!30BVv$[Q`V#F07$$ !3=x%>wUhxP\"F0$!3=x%>wUhxP#F07$$!3pY)='Hmd:8F0$!3#p%)='Hmd:BF07$$!3)H L(\\[OL^7F0$!3)HL(\\[OL^AF07$$!3IJI([U%[)=\"F0$!3IJI([U%[)=#F07$$!30^' p$pXnF6F0$!3$3lp$pXnF@F07$$!31*oHEfb,1\"F0$!3G*oHEfb,1#F07$$!3Ko-,!*y' [***!#=$!3%o-,!*y'[**>F07$$!3q-eI;*)4Z$*Ffu$!3F!eI;*)4Z$>F07$$!3Ua(H([ R7g()Ffu$!3WvH([R7g(=F07$$!3\"p=j(o_S=\")Ffu$!3p=j(o_S=\"=F07$$!3!p!3A ,!)f9vFfu$!3q!3A,!)f9v\"F07$$!3G5J[FaW$)oFfu$!3-6$[FaW$)o\"F07$$!3uWsY A%yjE'Ffu$!3[CnCUyjE;F07$$!37p#4]Xm.i&Ffu$!3#p#4]Xm.i:F07$$!3MRO+],=)* \\Ffu$!3%RO+],=)*\\\"F07$$!3#>>w()y/>O%Ffu$!3>>w()y/>O9F07$$!3;.#)y3\" )*3t$Ffu$!3K?)y3\")*3t8F07$$!3]iyp.&o5:$Ffu$!3D'yp.&o5:8F07$$!3Okp_U$= l[#Ffu$!3W'p_U$=l[7F07$$!3%)Hd@cn8#*=Ffu$!3)Hd@cn8#*=\"F07$$!3G&>LP,-% e7Ffu$!3`>LP,-%e7\"F07$$!3TrKYC+P=l!#>$!3sKYC+P=l5F07$$!3'************ ***f!#E$!30+++1+++5F07S7$$\"3'***************fFhz$\"30+++1+++5F07$$\"3 z1<#p$o9RlFbz$\"3;<#p$o9Rl5F07$$\"3XNUI,B)GA\"Ffu$\"3cB/8I#)GA6F07$$\" 353Xx$*eui=Ffu$\"3)3Xx$*eui=\"F07$$\"3^)H'[<4&o]#Ffu$\"3')H'[<4&o]7F07 $$\"3O5UXAY*y9$Ffu$\"3/@aCi%*y98F07$$\"39^bE'>CAu$Ffu$\"37bli>CAu8F07$ $\"3YjZ.X!=wN%Ffu$\"3NwM]/=wN9F07$$\"3O\")=wV#fS*\\Ffu$\"39)=wV#fS*\\ \"F07$$\"3i\"3$\\n$f%GcFfu$\"3;3$\\n$f%Gc\"F07$$\"3/lzVsy,\"G'Ffu$\"3S 'zVsy,\"G;F07$$\"3_;%)ye5&G](Ffu $\"3))H%**>5&G]F07$$\"3UF`\\wiL- 5F0$\"3kF`\\wiL-?F07$$\"3Ez2)QR5'f5F0$\"3Ez2)QR5'f?F07$$\"3FKD73QBE6F0 $\"3\\KD73QBE@F07$$\"3K'eH'=o?&=\"F0$\"3K'eH'=o?&=#F07$$\"3)3=vyb4*\\7 F0$\"3)3=vyb4*\\AF07$$\"3Nc>]m=_68F0$\"3Nc>]m=_6BF07$$\"3K%Q)*p%y!eP\" F0$\"3K%Q)*p%y!eP#F07$$\"3@:`+XC%[V\"F0$\"3*\\J0]WU[V#F07$$\"3ShHDM#>& )\\\"F0$\"3ShHDM#>&)\\#F07$$\"3+xcCA:mk:F0$\"3+xcCA:mkDF07$$\"3)G_!Qy& QAi\"F0$\"3)G_!Qy&QAi#F07$$\"3O`6QwLU%o\"F0$\"3e`6QwLU%o#F07$$\"33nE]d jm[F0$\"3@6.P8W%)RHF07$$\"3At*)*p@8 0+#F0$\"3At*)*p@80+$F07$$\"3y>%pV6!Hl?F0$\"3y>%pV6!HlIF07$$\"3hCq76w)R 7#F0$\"3hCq76w)R7$F07$$\"3O\"oB\"z%f\")=#F0$\"3O\"oB\"z%f\")=$F07$$\"3 O>z(e?S&[AF0$\"3O>z(e?S&[KF07$$\"3.*o^KYb;J#F0$\"3.*o^KYb;J$F07$$\"3ev Kvj@OtBF0$\"3evKvj@OtLF07$$\"39t!*\\gL'zV#F0$\"39t!*\\gL'zV$F07$$\"37O '**4*>=+DF0$\"37O'**4*>=+NF07$$\"3'3QAr_4Qc#F0$\"3'3QAr_4Qc$F07$$\"3uz 67&>5pi#F0$\"3uz67&>5pi$F07$$\"3!Q@Ic:$*[o#F0$\"3!Q@Ic:$*[o$F07$$\"3i. tur\"[8v#F0$\"3i.tur\"[8v$F07$$\"32F%y.L'y5GF0$\"32F%y.L'y5QF07$$\"3_! oEY!)fT(GF0$\"3_!oEY!)fT(QF07$$\"3Mn`v0j\"[$HF0$\"3Mn`v0j\"[$RF07$$\" \"$F*$\"\"%F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$F*F*Fhjl-F$6&7$7$Fhjl$! \"\"F*7$Fhjl$\"\"\"F*-%'SYMBOLG6$%'CIRCLEG\"#7Fajl-%&STYLEG6#%&POINTG- F$6&F[[m-Fc[m6$Fe[m\"#;FajlFg[m-%+AXESLABELSG6%Q\"x6\"Q!Fd\\m-%%FONTG6 #%(DEFAULTG-%*THICKNESSG6#\"\"#-%%VIEWG6$;F(F]jlFi\\m" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3 " }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The values of " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 32 " become progressi vely closer to " }{XPPEDIT 18 0 "-1" "6#,$\"\"\"!\"\"" }{TEXT -1 4 " a s " }{TEXT 274 1 "x" }{TEXT -1 67 " approaches 0 from the left (throug h values less than 0), so that: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Limit(f(x),x = 0^`-`) = -1;" "6#/-%&LimitG6$-%\"fG6#%\" xG/F*)\"\"!%\"-G,$\"\"\"!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 33 "On the other hand, the values of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 37 " become progressively closer to 1 as \+ " }{TEXT 275 1 "x" }{TEXT -1 71 " approaches 0 from the right (through values greater than 0), so that: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(f(x),x = 0^`+`) = 1;" "6#/-%&LimitG6$-%\"fG6#%\"xG/F*)\"\"!%\"+G\"\"\"" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT 261 4 "Note" }{TEXT -1 11 ": Although " } {XPPEDIT 18 0 "f(x)=x+abs(x)/x" "6#/-%\"fG6#%\"xG,&F'\"\"\"*&-%$absG6# F'F)F'!\"\"F)" }{TEXT -1 21 " is discontinuous at " }{XPPEDIT 18 0 "x= 0" "6#/%\"xG\"\"!" }{TEXT -1 49 ", it is continuous at all non-zero re al numbers. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2 " }}{PARA 0 "" 0 "" {TEXT -1 24 "The function f given by " }{XPPEDIT 18 0 "f(x) = x/(x-1);" "6#/-% \"fG6#%\"xG*&F'\"\"\",&F'F)F)!\"\"F+" }{TEXT -1 24 " has a discontinui ty at " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "As " } {TEXT 276 1 "x" }{TEXT -1 72 " approaches 1 from the left (through val ues less than 1), the numerator " }{TEXT 280 1 "x" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "f(x)=x/(x-1)" "6#/-%\"fG6#%\"xG*&F'\"\"\",&F'F)F)!\"\" F+" }{TEXT -1 37 " approaches 1, while the denominator " }{XPPEDIT 18 0 "x-1" "6#,&%\"xG\"\"\"F%!\"\"" }{TEXT -1 71 " approaches 0 from the \+ left (through negative values). It follows that " }{XPPEDIT 18 0 "f(x) " "6#-%\"fG6#%\"xG" }{TEXT -1 45 " takes progressively larger negative values (" }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 34 " appr oaches negative infinity) as " }{TEXT 278 1 "x" }{TEXT -1 14 " approac hes 1 " }{TEXT 261 13 "from the left" }{TEXT -1 11 ", so that: " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(f(x),x = 1^`-`) = -infinity;" "6#/-%&LimitG6$-%\"fG6#%\"xG/F*)\"\"\"%\"-G,$%)infinity G!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "This can be checked empirically by numerical calcula tions. For example, f(0.999)" }{XPPEDIT 18 0 "``=-999" "6#/%!G,$\"$*** !\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 29 "This means that the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 35 " approaches the vertical asymptote " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\" \"\"" }{TEXT -1 34 " \"going downwards from the left\". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "As " }{TEXT 281 1 " x" }{TEXT -1 14 " approaches 1 " }{TEXT 261 14 "from the right" } {TEXT -1 48 " (through values greater than 1), the numerator " }{TEXT 285 1 "x" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "f(x)=x/(x-1)" "6#/-%\"fG6 #%\"xG*&F'\"\"\",&F'F)F)!\"\"F+" }{TEXT -1 37 " approaches 1, while th e denominator " }{XPPEDIT 18 0 "x-1" "6#,&%\"xG\"\"\"F%!\"\"" }{TEXT -1 72 " approaches 0 from the right (through positive values). It foll ows that " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 45 " take s progressively larger positive values (" }{XPPEDIT 18 0 "f(x)" "6#-% \"fG6#%\"xG" }{TEXT -1 25 " approaches infinity) as " }{TEXT 283 1 "x " }{TEXT -1 14 " approaches 1 " }{TEXT 261 14 "from the right" }{TEXT -1 11 ", so that: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(f(x),x = 1^`+`) = infinity;" "6#/-%&LimitG6$-%\"fG6#%\"xG/F*)\" \"\"%\"+G%)infinityG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 72 "This can be checked empirically by nu merical calculations. For example, " }{XPPEDIT 18 0 "f(1.0001) = 10001 ;" "6#/-%\"fG6#-%&FloatG6$\"&,+\"!\"%F*" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 29 "This means that the graph of " }{XPPEDIT 18 0 "f(x )" "6#-%\"fG6#%\"xG" }{TEXT -1 35 " approaches the vertical asymptote \+ " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 33 " \"going upwards from the right\". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 13 "The graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 19 " also has the line " }{XPPEDIT 18 0 "y=1" "6#/%\"yG\"\" \"" }{TEXT -1 113 " as a horizontal asymptote, although this is not pe rtinent to the question of continuity. To see this note that" }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)=1+1/(x-1)" "6#/- %\"fG6#%\"xG,&\"\"\"F)*&F)F),&F'F)F)!\"\"F,F)" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "This expr ession for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 52 " ca n be obtained the following polynomial division. " }}{PARA 257 "" 0 " " {TEXT -1 27 " 1" }}{PARA 257 "" 0 "" {TEXT -1 24 " ______" }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "x-1" "6#,&%\"xG\"\"\"F%!\"\"" }{TEXT -1 5 " | " } {TEXT 286 1 "x" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 20 " \+ " }{XPPEDIT 18 0 "x-1" "6#,&%\"xG\"\"\"F%!\"\"" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 24 " _____" }} {PARA 257 "" 0 "" {TEXT -1 26 " 1" }}{PARA 0 " " 0 "" {TEXT -1 3 "As " }{TEXT 287 1 "x" }{TEXT -1 76 " gets progressi vely larger in the positive direction (approaches infinity), " } {XPPEDIT 18 0 "1/(x-1)" "6#*&\"\"\"F$,&%\"xGF$F$!\"\"F'" }{TEXT -1 40 " approaches 0 (from the right), so that " }{XPPEDIT 18 0 "f(x)=1+1/(x -1)" "6#/-%\"fG6#%\"xG,&\"\"\"F)*&F)F),&F'F)F)!\"\"F,F)" }{TEXT -1 43 " approaches 1 (from the right), in symbols " }{XPPEDIT 18 0 "f(x)->1 " "6#f*6#-%\"fG6#%\"xG7\"6$%)operatorG%&arrowG6\"\"\"\"F-F-F-" } {XPPEDIT 18 0 "``^`+`" "6#)%!G%\"+G" }{TEXT -1 5 ", as " }{XPPEDIT 18 0 "x->infinity" "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\"%)infinityGF*F* F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 3 "As " }{TEXT 288 1 "x" }{TEXT -1 85 " gets progressively \+ larger in the negative direction (approaches negative infinity), " } {XPPEDIT 18 0 "1/(x-1)" "6#*&\"\"\"F$,&%\"xGF$F$!\"\"F'" }{TEXT -1 39 " approaches 0 (from the left), so that " }{XPPEDIT 18 0 "f(x)=1+1/(x- 1)" "6#/-%\"fG6#%\"xG,&\"\"\"F)*&F)F),&F'F)F)!\"\"F,F)" }{TEXT -1 42 " approaches 1 (from the left), in symbols " }{XPPEDIT 18 0 "f(x)->1" " 6#f*6#-%\"fG6#%\"xG7\"6$%)operatorG%&arrowG6\"\"\"\"F-F-F-" }{XPPEDIT 18 0 "``^`-`;" "6#)%!G%\"-G" }{TEXT -1 5 ", as " }{XPPEDIT 18 0 "x->-i nfinity" "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\",$%)infinityG!\"\"F*F* F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "f := x -> x/(x-1):\n'f(x)'=f(x);\np1 := plot(f(x),x=-3..5,y=-3..5,discont=true, thickness=2):\np2 := plot([[[-3,1],[5,1]],[[1,-3],[1,5]]],color=black, linestyle=3):\nplots[display]([p1,p2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*&F'\"\"\",&F'F)F)!\"\"F+" }}{PARA 13 "" 1 "" {GLPLOT2D 339 306 306 {PLOTDATA 2 "6'-%'CURVESG6&7gn7$$!\"$\"\"!$\"3++ ++++++v!#=7$$!31;TY$Q6G\"H!#<$\"3B&\\$\\dGHWuF-7$$!3\"HC6W.\\p$GF1$\"3 !p*[k>Ew$R(F-7$$!3m#)eq))Qj^FF1$\"3mxIt.Z\\MtF-7$$!3s.Z$)=KvlEF1$\"3)3 H=%zw/ssF-7$$!3QXizC2G!e#F1$\"3=Wz/HH#p?(F-7$$!3+yZ:vqF-7$$!3KX@$=WDTL#F1$\"3$=\\\"GGFr +qF-7$$!3'=pD'e(Q&\\AF1$\"3(4PqG=SE#pF-7$$!3?qG#z&4`i@F1$\"3nT')H'**>F1$\"3mfG9[UCmm F-7$$!3`80j^5*H\">F1$\"3=)palR-rc'F-7$$!3zPdvJ\"3&H=F1$\"3OEReOo\"eY'F -7$$!3%oy-Fk(p`&ojF-7$$!39R8nQ;bj;F1$\"3'f;q)oVhXiF-7$$! 3Q()ey[h=(e\"F1$\"3o<&p+(ozMhF-7$$!3knCvH\\N)\\\"F1$\"3at&fW:mt*fF-7$$ !3]!3P!\\Us>9F1$\"3aWzL(o(HneF-7$$!3:>)***HRXL8F1$\"3,F$>p?2Xr&F-7$$!3 wVI7&=/8D\"F1$\"3af.KJ-8ebF-7$$!3A#G=Wa*el6F1$\"3I&[+=y>BQ&F-7$$!3S^j. Zn(o3\"F1$\"364+1j,:3_F-7$$!3xr.LhV(>+\"F1$\"3(=-.b.J\\+&F-7$$!3SiliRk %y8*F-$\"3q\\rvx;vuZF-7$$!3QtZWdB:q$)F-$\"3e)))o+,(QcXF-7$$!3!3!=-<<-T vF-$\"3=GWo1y2*H%F-7$$!3'yKt\\j[Wo'F-$\"38)=%)fo%R1SF-7$$!3!=JIi)*ek%e F-$\"35T)fq!>W*o$F-7$$!3E6lW74mN]F-$\"3Ed<5P]9\\LF-7$$!3D*)oz))ySNTF-$ \"32Oxj>ncDHF-7$$!3pNpn10\\ELF-$\"3'fG#fL*[h\\#F-7$$!3+/eIT&)ziCF-$\"3 m!Qfn**>h(>F-7$$!3?)383Dl,o\"F-$\"3e*o*R%Hx%Q9F-7$$!3)H*=jPMSX#)!#>$\" 3&>ExlJCth(Ffv7$$!3wsvuj')RY>!#?$\"365L`bvhU>F\\w7$$\"3K8*owyF2A)Ffv$! 3)=(4N/O1d*)Ffv7$$\"3;)3meyG[k\"F-$!3gDpPVcjo>F-7$$\"3TI2Cw!yh]#F-$!3, 2/#[jDVM$F-7$$\"3!4.jm\")fdL$F-$!3V_]Nv:Y0]F-7$$\"3!yZ!Rlp7%=%F-$!3K18 r)3BV>(F-7$$\"3BK^z0#pa-&F-$!3N\")*[C%)R-,\"F17$$\"3WR@IY`d)z&F-$!3e!H '*z-X,Q\"F17$$\"3AsjIGAk%o'F-$!3o.v6@dE;?F17$$\"3yR4X55:xuF-$!3)ehB%>F xjHF17$$\"3(\\!o^n18A$)F-$!3IQr`s'R*f\\F17$$\"3.R-\">M2ls)F-$!3#>.tQh? C&oF17$$\"33tOI;S)38*F-$!3%oU\\'=Wf]5!#;7$$\"3oaxA(*H;[$*F-$!3&eVHi&e7 M9F[[l7$$\"3GO=:y>Wl&*F-$!3JhZVy')=,AF[[l7$$\"3kxQho93u'*F-$!3EVQ.I9Do HF[[l7$$\"3)y\"f2f4s#y*F-$!3g%G89swB]%F[[l7$$\"3bQpI/2/P)*F-$!38kyOI<] OgF[[l7$$\"37ez`\\/O\"*)*F-$!3@BE+,4v/\"*F[[l7$$\"3\"zY`@K?&=**F-$!3kw Q>%*)*H<7!#:7$$\"3!)y*oZ>!oX**F-$!3r2=Zj\"\\4$=F_]l7$$\"3q*[%Qn+%G(**F -$!3Ogb.gU*=n$F_]l7$$\"2%******R********F1$!3KiT7Ummm;!\"*7gn7$$\"3%** *****4+++5F1$\"33ru2;+++5F_^l7$$\"3!=-kUlCF+\"F1$\"3!)o?@G/>!o$F_]l7$$ \"3VV!G&)H\\a+\"F1$\"3W8no\"e)4X=F_]l7$$\"32l?zUR<35F1$\"3#>!4\"G!)*RL 7F_]l7$$\"3$p3cqe)*3,\"F1$\"3]^#3uK,bF*F[[l7$$\"3?ITevyM;5F1$\"3xpW#ev -q@'F[[l7$$\"3pt@6krz@5F1$\"3iS2XoFv(o%F[[l7$$\"3Yg#o6u&pK5F1$\"3G$ebL J-&eJF[[l7$$\"3XZVA=VfV5F1$\"3KuBT5p(QR#F[[l7$$\"3?@lLs9Rl5F1$\"3hI(Q0 R^#H;F[[l7$$\"3'\\p[ki)=(3\"F1$\"3c$)Hv'eQpC\"F[[l7$$\"3K+s$4!)>^7\"F1 $\"3;_EM\"3SB**)F17$$\"3!fqDa(40j6F1$\"3;$G?'>80LrF17$$\"3I^d/@hO[7F1$ \"3**Qm3RSJE]F17$$\"3!)\\5$3zYUL\"F1$\"3!H$e()\\8!=*RF17$$\"3>NSy%G>(> 9F1$\"3I'Hx8![a#Q$F17$$\"3ie#>zAj*)\\\"F1$\"3&[bh`qbT+$F17$$\"3cgu/Td, \"e\"F1$\"3Ux,UlP7@FF17$$\"39!)>]nX(em\"F1$\"3+lc=VVy,DF17$$\"3%o%Qi]7 Y]#F17$$\"3k-9Qpb 59>F1$\"3qP)[yalR4#F17$$\"3v\\!*\\P,Q+?F1$\"3kd0&p+?'**>F17$$\"3qxCGd* 3q3#F1$\"3V**pp?c&*>>F17$$\"3AqP2x=\\q@F1$\"3ol;$GoTV&=F17$$\"3]540mBI YAF1$\"3:>\\\"o]tB!=F17$$\"3s7A**p$[kL#F1$\"39!=#)f(=D[0Oy*e C$F1$\"3g3eYGiDX9F17$$\"3?76Y%)Qr0S\"F17$$\"3'>&Q`=#f ke$F1$\"3'o,2a%*GmQ\"F17$$\"3eb\\mc4NnOF1$\"3sucS'z.\\P\"F17$$\"3i*p:J :?Pv$F1$\"3GWm72^9j8F17$$\"3-3n3#[$)>$QF1$\"3'QWxTS4JN\"F17$$\"3DNhqsf aQ$3O0)*RF1$\"3?QsZR(\\NL\"F17$$\"3q\"[%z% G2A3%F1$\"3>B%RC\"GWC8F17$$\"3IE@U&)G[kTF1$\"3bUk;(R2gJ\"F17$$\"3)eXtV \"yh]UF1$\"3tqv]FQj28F17$$\"3RFFL))fdLVF1$\"3)ztRE;y**H\"F17$$\"3]N17. FT=WF1$\"3'zQ*)>VLDH\"F17$$\"3\"p'p2Fpa-XF1$\"3UCj[Cl]&G\"F17$$\"3-d.0 Tv&)zXF1$\"3ofZuC2Mz7F17$$\"3'HH28A$[F1$\"3APr]Ce%4E\"F17$$\"3tysr2%)3 8\\F1$\"3qjuI@Ebb7F17$$\"\"&F*$\"3+++++++]7F1-%'COLOURG6&%$RGBG$\"*+++ +\"!\")$F*F*F_am-%*THICKNESSG6#\"\"#-F$6%7$7$F($\"\"\"F*7$Fd`mFham-Fi` m6&F[amF*F*F*-%*LINESTYLEG6#\"\"$-F$6%7$7$FhamF(7$FhamFd`mF[bmF]bm-%+A XESLABELSG6%Q\"x6\"Q\"yFjbm-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F(Fd`mFccm" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Cur ve 2" "Curve 3" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "When foll owing the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 51 " from left to right (or in taking sample values of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 4 " as " }{TEXT 289 1 "x" } {TEXT -1 27 " increases), the values of " }{XPPEDIT 18 0 "f(x)" "6#-% \"fG6#%\"xG" }{TEXT -1 63 " \"jump\" from large negative values to lar ge positive values as " }{TEXT 290 1 "x" }{TEXT -1 22 " increases thro ugh 1. " }}{PARA 0 "" 0 "" {TEXT -1 21 "Indeed the values of " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 33 " obtained by the \+ Maple procedure " }{TEXT 0 4 "plot" }{TEXT -1 26 " exhibit this phenom enon. " }}{PARA 0 "" 0 "" {TEXT -1 27 "The last point obtained by " } {TEXT 0 4 "plot" }{TEXT -1 4 " as " }{TEXT 291 1 "x" }{TEXT -1 60 " ap proaches 1 from the left and the first point obtained by " }{TEXT 0 4 "plot" }{TEXT -1 7 " after " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" } {TEXT -1 16 " are as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "plot(f(x),x=-3..5,y=-3..5,t hickness=2):\npts := op(1,op(1,%)):\nfor ct to nops(pts) do if pts[ct, 1]>1 then break end if end do:\npts[ct-1];\npts[ct];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"3gKLLLG^g**!#=$!3Equpa$oC_#!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"3I$3x19j:+\"!#<$\"3FO2CioP2k!#:" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "When option \"" } {TEXT 262 12 "discont=true" }{TEXT -1 15 "\" is used with " }{TEXT 0 4 "plot" }{TEXT -1 69 ", the two branches are plotted separately and t he last point with an " }{TEXT 292 1 "x" }{TEXT -1 60 " coordinate les s than 1 followed by the first point with an " }{TEXT 293 1 "x" } {TEXT -1 35 " coordinate greater than 1 are ... " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "plot(f(x),x =-3..5,y=-3..5,discont=true,thickness=2):\nop(1,ListTools[Reverse](op( 1,op(1,%))));\nop(1,op(2,op(1,%%)));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#7$$\"2%******R********!#<$!3KiT7Ummm;!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"3%*******4+++5!#<$\"33ru2;+++5!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "These points do not \+ appear in the previous graph because of the restriction placed on the \+ " }{TEXT 294 1 "y" }{TEXT -1 11 " variable. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 4 "Note" }{TEXT -1 11 ": Although \+ " }{XPPEDIT 18 0 "f(x) = x/(x-1);" "6#/-%\"fG6#%\"xG*&F'\"\"\",&F'F)F) !\"\"F+" }{TEXT -1 21 " is discontinuous at " }{XPPEDIT 18 0 "x=1" "6# /%\"xG\"\"\"" }{TEXT -1 46 ", it is continuous at all other real numbe rs. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 3 \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Consi der the function f defined by " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "f(x)=``" "6#/-%\"fG6#%\"xG%!G" }{TEXT -1 20 "the greate st integer" }{XPPEDIT 18 0 "``<=x" "6#1%!G%\"xG" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 61 "We could use double square brackets to de note this function. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 18 "Thus, for example," }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(Pi) = [[Pi]];" "6#/-%\"fG6#%#PiG7#7#F'" }{XPPEDIT 18 0 "``=3" "6#/%!G\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "f(sqrt(101 ))=[[sqrt(101)]]" "6#/-%\"fG6#-%%sqrtG6#\"$,\"7#7#-F(6#F*" }{XPPEDIT 18 0 "``=10" "6#/%!G\"#5" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "f(-Pi)=[[- Pi]]" "6#/-%\"fG6#,$%#PiG!\"\"7#7#,$F(F)" }{XPPEDIT 18 0 "``=-4" "6#/% !G,$\"\"%!\"\"" }{TEXT -1 1 "," }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "f(56+10^(-100))=[[56+10^(-100)]]" "6#/-%\"fG6#,&\"#c\" \"\")\"#5,$\"$+\"!\"\"F)7#7#,&F(F))F+,$F-F.F)" }{XPPEDIT 18 0 "``=56" "6#/%!G\"#c" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "f(56-10^(-100)) = [[56- 10^(-100)]];" "6#/-%\"fG6#,&\"#c\"\"\")\"#5,$\"$+\"!\"\"F.7#7#,&F(F))F +,$F-F.F." }{XPPEDIT 18 0 "``=55" "6#/%!G\"#b" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 42 "The value jumps up from one integer value " }{XPPEDIT 18 0 "n-1" "6#,&%\"nG\"\"\"F%!\"\"" }{TEXT -1 21 " to the next integer " }{TEXT 299 1 "n" }{TEXT -1 4 " as " }{TEXT 297 1 "x" } {TEXT -1 37 " increases through the integer value " }{TEXT 298 1 "n" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 33 "More precisely, for eac h integer " }{TEXT 300 1 "n" }{TEXT -1 1 "," }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)=[[x]]" "6#/-%\"fG6#%\"xG7#7#F'" } {XPPEDIT 18 0 "`` = n-1;" "6#/%!G,&%\"nG\"\"\"F'!\"\"" }{TEXT -1 7 " \+ for " }{XPPEDIT 18 0 "n-1<=x" "6#1,&%\"nG\"\"\"F&!\"\"%\"xG" } {XPPEDIT 18 0 "`` " 0 "" {MPLTEXT 1 0 289 "p1 := plot([[seq([i,i],i=-6..6)]$3],style=point,color=red,symbol= [circle,diamond,cross]):\np2 := plot([seq([i,i-1],i=-5..6)],style=poin t,color=red,symbol=circle):\np3 := plot(floor(x),x=-6..6,color=red,dis cont=true,thickness=2):\nplots[display]([p1,p2,p3],labels=[`x`,`y`],ti tle=`y = [[x]]`);" }}{PARA 13 "" 1 "" {GLPLOT2D 476 476 476 {PLOTDATA 2 "6+-%'CURVESG6&7/7$$!\"'\"\"!F(7$$!\"&F*F,7$$!\"%F*F/7$$!\"$F*F27$$! \"#F*F57$$!\"\"F*F87$$F*F*F;7$$\"\"\"F*F=7$$\"\"#F*F@7$$\"\"$F*FC7$$\" \"%F*FF7$$\"\"&F*FI7$$\"\"'F*FL-%'SYMBOLG6#%'CIRCLEG-%'COLOURG6&%$RGBG $\"*++++\"!\")F;F;-%&STYLEG6#%&POINTG-F$6&F&-FO6#%(DIAMONDGFRFY-F$6&F& -FO6#%&CROSSGFRFY-F$6&7.7$F,F(7$F/F,7$F2F/7$F5F27$F8F57$F;F87$F=F;7$F@ F=7$FCF@7$FFFC7$FIFF7$FLFIFRFNFY-F$607S7$$!3!******z)******f!#vCP#ffFfpF(7$$!3!)HNdh%3z$fFfpF(7$$!349))e%HQk \"fFfpF(7$$!3!=)*f:KFj!z&FfpF(7$$!3amz 1.OZrdFfpF(7$$!3O!G-:'\\!*\\dFfpF(7$$!3e`?-dxCGdFfpF(7$$!3<^H_FqP2dFfp F(7$$!3EZho0WU)o&FfpF(7$$!3cc`=0z)em&FfpF(7$$!3!*\\N9LlzYcFfpF(7$$!3$3 ([))G()eCcFfpF(7$$!3Y@$[\"f5$\\g&FfpF(7$$!3iwU()zMO$e&FfpF(7$$!33vr6Wg #Gc&FfpF(7$$!3oHJU%Q(RTbFfpF(7$$!3>1/_&=><_&FfpF(7$$!3E([r&*e$\\+bFfpF (7$$!3iDcsghWyaFfpF(7$$!3e3H54QDfaFfpF(7$$!34?(eNaD&QaFfpF(7$$!3tI$*)p @6rT&FfpF(7$$!3/D@Cw9;'R&FfpF(7$$!3W.O&[_\"*eP&FfpF(7$$!3gb(o(*>&Q``Ff pF(7$$!3Bs2nlA;L`FfpF(7$$!3V?B-n*p:J&FfpF(7$$!3[M-?NT+#H&FfpF(7$$!3ev_ +8Nhq_FfpF(7$$!3G))*HXg'[]_FfpF(7$$!3+W#oe=[%H_FfpF(7$$!3WkN:'Gz)3_Ffp F(7$$!3-2(yVbXt=&FfpF(7$$!3#*)yM8,1m;&FfpF(7$$!3!)4Q4LoRX^FfpF(7$$!3(y %pbxKOC^FfpF(7$$!3$[9zWiN]5&FfpF(7$$!3K_u#H&R)G3&FfpF(7$$!3)Ri>QBrI1&F fpF(7$$!3Ey#oGuY>/&FfpF(7$$!3bINg4zs@]FfpF(7$$!35+++7+++]FfpF(7S7$$!31 +++!*******\\FfpF,7$$!3_w#pi$G?y\\FfpF,7$$!3o@pN\\sBf\\FfpF,7$$!3?k^Kj %3z$\\FfpF,7$$!3'pcaiHQk\"\\FfpF,7$$!3[j-9t,2&*[FfpF,7$$!3q\\Jq(=f_([F fpF,7$$!37z+yfgua[FfpF,7$$!3Y1x``8`L[FfpF,7$$!3]1t7$o%Q7[FfpF,7$$!3u6S NLFj!z%FfpF,7$$!3YhQ:/OZrZFfpF,7$$!3K+>]i\\!*\\ZFfpF,7$$!3/X]$zvZ#GZFf pF,7$$!3>fCNGqP2ZFfpF,7$$!3![%)RkSC%)o%FfpF,7$$!3W3*[e!z)em%FfpF,7$$!3 ?O2tLlzYYFfpF,7$$!32DKQH()eCYFfpF,7$$!3FX!o&f5$\\g%FfpF,7$$!3wIx?![jLe %FfpF,7$$!3%*y%oV/EGc%FfpF,7$$!3+>()e%Q(RTXFfpF,7$$!3O$G2c=><_%FfpF,7$ $!3ohMd*e$\\+XFfpF,7$$!3;5%R1;Y%yWFfpF,7$$!3SB*R*3QDfWFfpF,7$$!3cAGJVb _QWFfpF,7$$!3^vxl;76&Q`VFfpF,7$$!3h@M+lA;LVFfpF,7$$!3n+'oi'*p:J%FfpF,7$$!37^#oV8/ ?H%FfpF,7$$!3aHx37NhqUFfpF,7$$!33N>`.m[]UFfpF,7$$!3zOgy%=[%HUFfpF,7$$! 3\"33*)\\Gz)3UFfpF,7$$!3w)3GJbXt=%FfpF,7$$!3+87+5ggmTFfpF,7$$!3'oRv;$o RXTFfpF,7$$!3M,W0wKOCTFfpF,7$$!3p(G**GiN]5%FfpF,7$$!3%y)*e7&R)G3%FfpF, 7$$!374>2K72jSFfpF,7$$!3mlg.Tn%>/%FfpF,7$$!3>U/p2zs@SFfpF,7$$!3%****** *4+++SFfpF,7S7$$!3y*****>*******RFfpF/7$$!3w(3#=QG?yRFfpF/7$$!3[qQ>^sB fRFfpF/7$$!3q(zw]Y3z$RFfpF/7$$!3G?.#zHQk\"RFfpF/7$$!3QV0su,2&*QFfpF/7$ $!3)e=/#*=f_(QFfpF/7$$!3si!*>hguaQFfpF/7$$!3FJ=([NJN$QFfpF/7$$!3vWoP%o %Q7QFfpF/7$$!3qUl^MFj!z$FfpF/7$$!3]b(R_gt9x$FfpF/7$$!3E?:]j\\!*\\PFfpF /7$$!3hN![)exCGPFfpF/7$$!3@n>=HqP2PFfpF/7$$!3!Ga$>2WU)o$FfpF/7$$!3))fC ^1z)em$FfpF/7$$!31AzJMlzYOFfpF/7$$!3@!e\"))H()eCOFfpF/7$$!3`px)*f5$\\g $FfpF/7$$!3/%=T0[jLe$FfpF/7$$!3E$y>Y/EGc$FfpF/7$$!3K3Vv%Q(RTNFfpF/7$$! 3lfTp&=><_$FfpF/7$$!35Oad*e$\\+NFfpF/7$$!3o%>`0;Y%yMFfpF/7$$!35Rpx3QDf MFfpF/7$$!3fCp1Vb_QMFfpF/7$$!3s?iK;76&Q`LFfpF/7$$!3aqgLkA;LLFfpF/7$$!3[!)[^l*p: J$FfpF/7$$!3Ini`LT+#H$FfpF/7$$!3'R=q6^81F$FfpF/7$$!3+\")Q`-m[]KFfpF/7$ $!38HQq$=[%HKFfpF/7$$!33)fCQGz)3KFfpF/7$$!3Oru(=bXt=$FfpF/7$$!33Pwm3gg mJFfpF/7$$!3O%)pDIoRXJFfpF/7$$!3Pa=buKOCJFfpF/7$$!37I%>8iN]5$FfpF/7$$! 3!Q_!f\\R)G3$FfpF/7$$!3#Q>C.BrI1$FfpF/7$$!3<_Q?Rn%>/$FfpF/7$$!3%QNxd!z s@IFfpF/7$$!3A+++3+++IFfpF/7S7$$!3%******R*******HFfpF27$$!3+**[4SG?yH FfpF27$$!3I>3.`sBfHFfpF27$$!3lJ%GoY3z$HFfpF27$$!3ftge*HQk\"HFfpF27$$!3 iC3Iw,2&*GFfpF27$$!3'HA02>f_(GFfpF27$$!3?Z!=E1YZ&GFfpF27$$!3(p&f?c8`LG FfpF27$$!3*QQEco%Q7GFfpF27$$!3mt!zctK1z#FfpF27$$!3)*\\cK1OZrFFfpF27$$! 3?S6]k\\!*\\FFfpF27$$!3iE5wfxCGFFfpF27$$!3nv9,IqP2FFfpF27$$!3OSs%zSC%) o#FfpF27$$!3w6g<2z)em#FfpF27$$!3O3^!\\`'zYEFfpF27$$!3MN*z.t)eCEFfpF27$ $!3y$\\2/1J\\g#FfpF27$$!3=QY(3[jLe#FfpF27$$!3c(3r[/EGc#FfpF27$$!33)*)> \\Q(RTDFfpF27$$!3QO5y&=><_#FfpF27$$!335ud*e$\\+DFfpF27$$!3?zpYghWyCFfp F27$$!3QaRh3QDfCFfpF27$$!3iE5#GaD&QCFfpF27$$!3\\lY*f@6rT#FfpF27$$!3Cig *\\ZhhR#FfpF27$$!3'=IkL_\"*eP#FfpF27$$!3ix$4!)>&Q`BFfpF27$$!3[>(oOEiJL #FfpF27$$!3Gg6wk*p:J#FfpF27$$!3%RG/F8/?H#FfpF27$$!3#zj_-^81F#FfpF27$$! 3!y#e`,m[]AFfpF27$$!3#>i@E=[%HAFfpF27$$!3N:,m#Gz)3AFfpF27$$!3a`oi]bM(= #FfpF27$$!3;hSL2ggm@FfpF27$$!3Ur&Q)GoRX@FfpF27$$!3R2$\\IFjV7#FfpF27$$! 3as&R(>c.0@FfpF27$$!3Mf?#z%R)G3#FfpF27$$!3_ykdG72j?FfpF27$$!37R;PPn%>/ #FfpF27$$!3/lU'Q!zs@?FfpF27$$!30+++1+++?FfpF27S7$$!3))*****f*******>Ff pF57$$!3Y5x+UG?y>FfpF57$$!3cox'[DP#f>FfpF57$$!3Ql+eo%3z$>FfpF57$$!3!p# =D,$Qk\">FfpF57$$!3?06)yf_(=FfpF57$$!3YJq.kgua=Ff pF57$$!3X#3SvNJN$=FfpF57$$!3OAf(oo%Q7=FfpF57$$!3i/;%otK1z\"FfpF57$$!3[ W:T2OZr<_\"FfpF57$$!3]%Qz&* e$\\+:FfpF57$$!3;k2QghWy9FfpF57$$!3kp4X3QDf9FfpF57$$!35H^dUb_Q9FfpF57$ $!3F5Jm:76<9FfpF57$$!3E32eu9;'R\"FfpF57$$!3'y'y'G_\"*eP\"FfpF57$$!3_=H U(>&Q`8FfpF57$$!3ko8+jA;L8FfpF57$$!3ISu+k*p:J\"FfpF57$$!3M+B(=8/?H\"Ff pF57$$!3)=4N$4Nhq7FfpF57$$!3;ux`+m[]7FfpF57$$!3r9%R:=[%H7FfpF57$$!3iKc \\\"Gz)37FfpF57$$!3qNiP\\bM(=\"FfpF57$$!3C&[+g+1m;\"FfpF57$$!3#*e,UFoR X6FfpF57$$!3TgnarKOC6FfpF57$$!3'\\rf\"=c.06FfpF57$$!3I&f`i%R)G3\"FfpF5 7$$!3kj(GoArI1\"FfpF57$$!32E%RbtY>/\"FfpF57$$!3\"p<^>!zs@5FfpF57$$!36+ ++/+++5FfpF57S7$$!2%******z********FfpF87$$!3E>_?R%G?y*!#=F87$$!3)ff_()F^_nF87$$!3Cd,ca1YZ&)F^_nF87$$!3<_F^_nF87$$!3A*e8e*e$\\+&F^_nF87$$! 3s'[XHghWy%F^_nF87$$!3'y%)zG3QDf%F^_nF87$$!3K6BHBaD&Q%F^_nF87$$!3_^bJ` @6rTF^_nF87$$!3wSNlTZhhRF^_nF87$$!3iQVrB_\"*ePF^_nF87$$!3#>fk$o>&Q`$F^ _nF87$$!3*o F87$$!3VX[53D72jF`gnF87$$!3U-8sqLn%>%F`gnF87$$!3EL)3Q+!zs@F`gnF87$$!37 +++++++?!#EF87S7$$\"37+++++++?F]hnF;7$$\"3s/y%zg:(z@F`gnF;7$$\"3_O#G&H VFwSF`gnF;7$$\"3Lb+$o'H:4iF`gnF;7$$\"33))>C3(phN)F`gnF;7$$\"3]ShQ0#)H \\5F^_nF;7$$\"3EMq#H13uC\"F^_nF;7$$\"3[U)RaMRDX\"F^_nF;7$$\"3+@zD6kok; F^_nF;7$$\"3a(QX(=J:w=F^_nF;7$$\"3yU'e*>En$4#F^_nF;7$$\"3+6c-:RE&G#F^_ nF;7$$\"3o)>'*\\L]4]#F^_nF;7$$\"355*HTQAvr#F^_nF;7$$\"3E\"3&H$oHi#HF^_ nF;7$$\"3nUOX0fv:JF^_nF;7$$\"3y\\)o\\\"47TLF^_nF;7$$\"3*[>0#RY.KNF^_nF ;7$$\"3))[NB'o7Tv$F^_nF;7$$\"3Uv0`(Q*o]RF^_nF;7$$\"3_RXe%=lj;%F^_nF;7$ $\"31UIEY&RY2aF^_nF;7$$\"3n )o2ndWZh&F^_nF;7$$\"3[[WoYy))GeF^_nF;7$$\"3BfkMe_QQgF^_nF;7$$\"3QhcGwZ 3TiF^_nF;7$$\"333ajJ![hY'F^_nF;7$$\"35B)fmPx$omF^_nF;7$$\"3%*)ziuO+V)o F^_nF;7$$\"3QJof*oe*zqF^_nF;7$$\"3cSX#e\"\\'QH(F^_nF;7$$\"3+(z-Y+M^\\( F^_nF;7$$\"3EFzU&>=bq(F^_nF;7$$\"3Z/&)o'>27\"zF^_nF;7$$\"3A@Qu=XaE\")F ^_nF;7$$\"3w24L`*RRL)F^_nF;7$$\"3GSD)*R<.Y&)F^_nF;7$$\"3_kyb*Hnjv)F^_n F;7$$\"31D9?MQk\\*)F^_nF;7$$\"3!=p[Tbg6<*F^_nF;7$$\"3Y:&*=\\xGp$*F^_nF ;7$$\"3wpy#HmK0e*F^_nF;7$$\"3o;\">'**4s#y*F^_nF;7$$\"2%******z******** FfpF;7S7$$\"36+++/+++5FfpF=7$$\"3a*G#*z:(z@5FfpF=7$$\"3WJA8XFwS5FfpF=7 $$\"3iM*>9`\"4i5FfpF=7$$\"35t\"[()phN3\"FfpF=7$$\"3z%*)=@#)H\\5\"FfpF= 7$$\"3USPz23uC6FfpF=7$$\"3aoH'f$RDX6FfpF=7$$\"3a<*fCkok;\"FfpF=7$$\"3k xS78`h(=\"FfpF=7$$\"3Q&ReJEn$47FfpF=7$$\"3`b%)e#RE&G7FfpF=7$$\"3')R#* \\M]4]7FfpF=7$$\"39#)fKRAvr7FfpF=7$$\"3I;!f\"pHi#H\"FfpF=7$$\"34i!*H\" fv:J\"FfpF=7$$\"3eO/;#47TL\"FfpF=7$$\"3y0x]kM?`8FfpF=7$$\"3'*4<7p7Tv8F fpF=7$$\"3u\"ys\"R*o]R\"FfpF=7$$\"3)y!>z=lj;9FfpF=7$$\"3N3w([&R'RQb@:FfpF=7$$\"3OI!\\:>Y2a\"FfpF=7$$\"3!4([UdWZh:FfpF=7$$\"3 u*)oL%y))Ge\"FfpF=7$$\"3t\"H>a_QQg\"FfpF=7$$\"39K@8x%3Ti\"FfpF=7$$\"3[ \"3xD![hY;FfpF=7$$\"3OJ')*ptPom\"FfpF=7$$\"3ofD*f.I%)o\"FfpF=7$$\"3m*p F\"oe*zq\"FfpF=7$$\"383\\m!\\'QHFfpF=7$$\"3NO7FfpF=7$$\"3#RdgWE`!e>FfpF=7$$\"35B )[!)4s#y>FfpF=7$$\"3))*****f*******>FfpF=7S7$$\"30+++1+++?FfpF@7$$\"3+ ,^!*frz@?FfpF@7$$\"3q!=ppui2/#FfpF@7$$\"3Mo:QPRDX@FfpF @7$$\"3-VSzV'ok;#FfpF@7$$\"35;OP9`h(=#FfpF@7$$\"3ME4KksO4AFfpF@7$$\"3, ]Vn$RE&GAFfpF@7$$\"3#)f))\\N]4]AFfpF@7$$\"3Qt*Q-C_lj;CFfpF@ 7$$\"3W7*G^&R5!3:EgeCFfpF@7$$\"3ij*=U\"3GyCFfpF@7$$\"3 #**eA/T1&*\\#FfpF@7$$\"3\"3-L&RQb@DFfpF@7$$\"3jXgQ\">Y2a#FfpF@7$$\"3Qt *yrXu9c#FfpF@7$$\"3]M`+%y))Ge#FfpF@7$$\"3vPR+D&QQg#FfpF@7$$\"38)pNmZ3T i#FfpF@7$$\"3QA1*>![hYEFfpF@7$$\"3_!GJjtPom#FfpF@7$$\"3sR)Q_.I%)o#FfpF @7$$\"31;dHne*zq#FfpF@7$$\"34itu*['QHFFfpF@7$$\"3?sTY)R8&\\FFfpF@7$$\" 33y$yt\"=bqFFfpF@7$$\"3k%))Rtr?6z#FfpF@7$$\"3YYJP\\Wl7GFfpF@7$$\"3$)Qf m#*RRLGFfpF@7$$\"3eG9;rJgaGFfpF@7$$\"3g#p]psOc(GFfpF@7$$\"3YF/E!Qk\\*G FfpF@7$$\"3nSz2_g6:TAvrKFfpFC7$$ \"3yK!=3(Hi#H$FfpFC7$$\"3>dk!Gfv:J$FfpFC7$$\"37Sv[$47TL$FfpFC7$$\"3%z2 #olM?`LFfpFC7$$\"3y>%=,F6aP$FfpFC7$$\"3YIA,S*o]R$FfpFC7$$\"3(f\")e%>lj ;MFfpFC7$$\"3v;-QbRY2a$FfpFC7$$\"3S vI$pXu9c$FfpFC7$$\"3GzPn$y))Ge$FfpFC7$$\"3y$e)eC&QQg$FfpFC7$$\"3dk#RhZ 3Ti$FfpFC7$$\"31jTS,[hYOFfpFC7$$\"3WHRmNx$om$FfpFC7$$\"3_>^[M+V)o$FfpF C7$$\"3pKPYme*zq$FfpFC7$$\"30;)H))['QHPFfpFC7$$\"3+>hY(R8&\\PFfpFC7$$ \"3'3<'H;=bqPFfpFC7$$\"3#>Svhr?6z$FfpFC7$$\"3jGD7[Wl7QFfpFC7$$\"3\"HOK 8*RRLQFfpFC7$$\"3k:IupJgaQFfpFC7$$\"3kX\"[asOc(QFfpFC7$$\"3))p0oyV'\\* QFfpFC7$$\"3>w%4/0;r\"RFfpFC7$$\"3=1enp(Gp$RFfpFC7$$\"3%y9'zgK0eRFfpFC 7$$\"3;YEA%4s#yRFfpFC7$$\"3y*****>*******RFfpFC7S7$$\"3%*******4+++SFf pFF7$$\"3\\B2tjrz@SFfpFF7$$\"3LyIk]FwSSFfpFF7$$\"3!e$[nO:4iSFfpFF7$$\" 3/Lau. -%RDXTFfpFF7$$\"3_$Hikkok;%FfpFF7$$\"3]$psoJ:w=%FfpFF7$$\"3E))fkmsO4UF fpFF7$$\"3aQh%eRE&GUFfpFF7$$\"3q*4)\\P]4]UFfpFF7$$\"3'\\&\\1UAvrUFfpFF 7$$\"3!3aZ;(Hi#H%FfpFF7$$\"3>b,c$fv:J%FfpFF7$$\"3b\"4^T47TL%FfpFF7$$\" 3!QEpiY.KN%FfpFF7$$\"3#\\x;1F6aP%FfpFF7$$\"3sa>VS*o]R%FfpFF7$$\"3BpAz> lj;WFfpFF7$$\"31@:jbRY2a%FfpFF7$$ \"3WxrocWZhXFfpFF7$$\"3]CAM$y))Ge%FfpFF7$$\"3\")HK4,:27\"z%FfpFF7$$\"3C6>(oWaE\"[FfpFF 7$$\"3+(y)***)RRL[FfpFF7$$\"39.YKoJga[FfpFF7$$\"3m)fXRsOc([FfpFF7$$\"3 I725xV'\\*[FfpFF7$$\"3;75u[g6<\\FfpFF7$$\"3)34GzwGp$\\FfpFF7$$\"3MMR'* eK0e\\FfpFF7$$\"3#yb4B4s#y\\FfpFF7$$\"31+++!*******\\FfpFF7S7$$\"35+++ 7+++]FfpFI7$$\"3tMNklrz@]FfpFI7$$\"3fF+[_FwS]FfpFI7$$\"3?qkUQ:4i]FfpFI 7$$\"3!f=6aqhN3&FfpFI7$$\"3>=+WG)H\\5&FfpFI7$$\"3#y)yz83uC^FfpFI7$$\"3 #e!*Q;%RDX^FfpFI7$$\"3A>kzZ'ok;&FfpFI7$$\"3kKA7=`h(=&FfpFI7$$\"3x=&3yE n$4_FfpFI7$$\"3ZL?$pRE&G_FfpFI7$$\"3k>x\\Q]4]_FfpFI7$$\"3UYz(HC_Y2a&FfpF I7$$\"3#*z7WcWZhbFfpFI7$$\"3Ep1,$y))Ge&FfpFI7$$\"3&\\(yvB&QQg&FfpFI7$$ \"3c'RY^Z3Ti&FfpFI7$$\"3SW7B+[hYcFfpFI7$$\"3wF#HVtPom&FfpFI7$$\"3czw(H .I%)o&FfpFI7$$\"3^l(*zke*zq&FfpFI7$$\"3TCZ*p['QHdFfpFI7$$\"3s6+Z&R8&\\ dFfpFI7$$\"3+c<89=bqdFfpFI7$$\"3cNk%Qr?6z&FfpFI7$$\"3(HH@cWaE\"eFfpFI7 $$\"336_m))RRLeFfpFI7$$\"3?!>1p;.Y&eFfpFI7$$\"37_IWAnjveFfpFI7$$\"3;b3 _vV'\\*eFfpFI7$$\"3oZD2Zg6 " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Consider the function f d efined by " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = \+ (-1)^[[x]];" "6#/-%\"fG6#%\"xG),$\"\"\"!\"\"7#7#F'" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " jumps from 1 to \+ " }{XPPEDIT 18 0 "-1;" "6#,$\"\"\"!\"\"" }{TEXT -1 4 " as " }{TEXT 302 1 "x" }{TEXT -1 42 " increases through each odd integer, and " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 12 " jumps from " } {XPPEDIT 18 0 "-1" "6#,$\"\"\"!\"\"" }{TEXT -1 9 " to 1 as " }{TEXT 303 1 "x" }{TEXT -1 37 " increases through each even integer." }} {PARA 0 "" 0 "" {TEXT -1 16 "More precisely, " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)=PIECEWISE([1, `if`* n<=x and x " 0 "" {MPLTEXT 1 0 360 "p1 : = plot([[seq([2*i,1],i=-3..3),seq([2*i-1,-1],i=-2..3)]$3],style=point, \n color=red,symbol=[circle,diamond,cross]):\np2 := plot([seq([2*i-1, 1],i=-2..3),seq([2*i,-1],i=-2..3)],style=point,\n color=red,symbol=ci rcle):\np3 := plot((-1)^floor(x),x=-6..6,color=red,discont=true,thickn ess=2):\nplots[display]([p1,p2,p3],labels=[`x`,`y`],view=-1.9..1.9,yti ckmarks=3);" }}{PARA 13 "" 1 "" {GLPLOT2D 579 184 184 {PLOTDATA 2 "6+- %'CURVESG6&7/7$$!\"'\"\"!$\"\"\"F*7$$!\"%F*F+7$$!\"#F*F+7$$F*F*F+7$$\" \"#F*F+7$$\"\"%F*F+7$$\"\"'F*F+7$$!\"&F*$!\"\"F*7$$!\"$F*FA7$FAFA7$F+F A7$$\"\"$F*FA7$$\"\"&F*FA-%'SYMBOLG6#%'CIRCLEG-%'COLOURG6&%$RGBG$\"*++ ++\"!\")F4F4-%&STYLEG6#%&POINTG-F$6&F&-FO6#%(DIAMONDGFRFY-F$6&F&-FO6#% &CROSSGFRFY-F$6&7.7$F?F+7$FDF+7$FAF+7$F+F+7$FIF+7$FLF+7$F.FA7$F1FA7$F4 FA7$F6FA7$F9FA7$FvCP#ffFfpF+7$$!3!)HNdh%3z$fFfpF+7$$!349))e%HQk\"fFfpF +7$$!3!=)*f:KFj!z&FfpF+7$$!3amz1.OZrdF fpF+7$$!3O!G-:'\\!*\\dFfpF+7$$!3e`?-dxCGdFfpF+7$$!3<^H_FqP2dFfpF+7$$!3 EZho0WU)o&FfpF+7$$!3cc`=0z)em&FfpF+7$$!3!*\\N9LlzYcFfpF+7$$!3$3([))G() eCcFfpF+7$$!3Y@$[\"f5$\\g&FfpF+7$$!3iwU()zMO$e&FfpF+7$$!33vr6Wg#Gc&Ffp F+7$$!3oHJU%Q(RTbFfpF+7$$!3>1/_&=><_&FfpF+7$$!3E([r&*e$\\+bFfpF+7$$!3i DcsghWyaFfpF+7$$!3e3H54QDfaFfpF+7$$!34?(eNaD&QaFfpF+7$$!3tI$*)p@6rT&Ff pF+7$$!3/D@Cw9;'R&FfpF+7$$!3W.O&[_\"*eP&FfpF+7$$!3gb(o(*>&Q``FfpF+7$$! 3Bs2nlA;L`FfpF+7$$!3V?B-n*p:J&FfpF+7$$!3[M-?NT+#H&FfpF+7$$!3ev_+8Nhq_F fpF+7$$!3G))*HXg'[]_FfpF+7$$!3+W#oe=[%H_FfpF+7$$!3WkN:'Gz)3_FfpF+7$$!3 -2(yVbXt=&FfpF+7$$!3#*)yM8,1m;&FfpF+7$$!3!)4Q4LoRX^FfpF+7$$!3(y%pbxKOC ^FfpF+7$$!3$[9zWiN]5&FfpF+7$$!3K_u#H&R)G3&FfpF+7$$!3)Ri>QBrI1&FfpF+7$$ !3Ey#oGuY>/&FfpF+7$$!3bINg4zs@]FfpF+7$$!35+++7+++]FfpF+7S7$$!31+++!*** ****\\FfpFA7$$!3_w#pi$G?y\\FfpFA7$$!3o@pN\\sBf\\FfpFA7$$!3?k^Kj%3z$\\F fpFA7$$!3'pcaiHQk\"\\FfpFA7$$!3[j-9t,2&*[FfpFA7$$!3q\\Jq(=f_([FfpFA7$$ !37z+yfgua[FfpFA7$$!3Y1x``8`L[FfpFA7$$!3]1t7$o%Q7[FfpFA7$$!3u6SNLFj!z% FfpFA7$$!3YhQ:/OZrZFfpFA7$$!3K+>]i\\!*\\ZFfpFA7$$!3/X]$zvZ#GZFfpFA7$$! 3>fCNGqP2ZFfpFA7$$!3![%)RkSC%)o%FfpFA7$$!3W3*[e!z)em%FfpFA7$$!3?O2tLlz YYFfpFA7$$!32DKQH()eCYFfpFA7$$!3FX!o&f5$\\g%FfpFA7$$!3wIx?![jLe%FfpFA7 $$!3%*y%oV/EGc%FfpFA7$$!3+>()e%Q(RTXFfpFA7$$!3O$G2c=><_%FfpFA7$$!3ohMd *e$\\+XFfpFA7$$!3;5%R1;Y%yWFfpFA7$$!3SB*R*3QDfWFfpFA7$$!3cAGJVb_QWFfpF A7$$!3^vxl;76&Q`VFfpFA7$$!3h@M+lA;LVFfpFA7$$!3n+'oi'*p:J%FfpFA7$$!37^#oV8/?H%FfpF A7$$!3aHx37NhqUFfpFA7$$!33N>`.m[]UFfpFA7$$!3zOgy%=[%HUFfpFA7$$!3\"33*) \\Gz)3UFfpFA7$$!3w)3GJbXt=%FfpFA7$$!3+87+5ggmTFfpFA7$$!3'oRv;$oRXTFfpF A7$$!3M,W0wKOCTFfpFA7$$!3p(G**GiN]5%FfpFA7$$!3%y)*e7&R)G3%FfpFA7$$!374 >2K72jSFfpFA7$$!3mlg.Tn%>/%FfpFA7$$!3>U/p2zs@SFfpFA7$$!3%*******4+++SF fpFA7S7$$!3y*****>*******RFfpF+7$$!3w(3#=QG?yRFfpF+7$$!3[qQ>^sBfRFfpF+ 7$$!3q(zw]Y3z$RFfpF+7$$!3G?.#zHQk\"RFfpF+7$$!3QV0su,2&*QFfpF+7$$!3)e=/ #*=f_(QFfpF+7$$!3si!*>hguaQFfpF+7$$!3FJ=([NJN$QFfpF+7$$!3vWoP%o%Q7QFfp F+7$$!3qUl^MFj!z$FfpF+7$$!3]b(R_gt9x$FfpF+7$$!3E?:]j\\!*\\PFfpF+7$$!3h N![)exCGPFfpF+7$$!3@n>=HqP2PFfpF+7$$!3!Ga$>2WU)o$FfpF+7$$!3))fC^1z)em$ FfpF+7$$!31AzJMlzYOFfpF+7$$!3@!e\"))H()eCOFfpF+7$$!3`px)*f5$\\g$FfpF+7 $$!3/%=T0[jLe$FfpF+7$$!3E$y>Y/EGc$FfpF+7$$!3K3Vv%Q(RTNFfpF+7$$!3lfTp&= ><_$FfpF+7$$!35Oad*e$\\+NFfpF+7$$!3o%>`0;Y%yMFfpF+7$$!35Rpx3QDfMFfpF+7 $$!3fCp1Vb_QMFfpF+7$$!3s?iK;76&Q`LFfpF+7$$!3aqgLkA;LLFfpF+7$$!3[!)[^l*p:J$FfpF+ 7$$!3Ini`LT+#H$FfpF+7$$!3'R=q6^81F$FfpF+7$$!3+\")Q`-m[]KFfpF+7$$!38HQq $=[%HKFfpF+7$$!33)fCQGz)3KFfpF+7$$!3Oru(=bXt=$FfpF+7$$!33Pwm3ggmJFfpF+ 7$$!3O%)pDIoRXJFfpF+7$$!3Pa=buKOCJFfpF+7$$!37I%>8iN]5$FfpF+7$$!3!Q_!f \\R)G3$FfpF+7$$!3#Q>C.BrI1$FfpF+7$$!3<_Q?Rn%>/$FfpF+7$$!3%QNxd!zs@IFfp F+7$$!3A+++3+++IFfpF+7S7$$!3%******R*******HFfpFA7$$!3+**[4SG?yHFfpFA7 $$!3I>3.`sBfHFfpFA7$$!3lJ%GoY3z$HFfpFA7$$!3ftge*HQk\"HFfpFA7$$!3iC3Iw, 2&*GFfpFA7$$!3'HA02>f_(GFfpFA7$$!3?Z!=E1YZ&GFfpFA7$$!3(p&f?c8`LGFfpFA7 $$!3*QQEco%Q7GFfpFA7$$!3mt!zctK1z#FfpFA7$$!3)*\\cK1OZrFFfpFA7$$!3?S6]k \\!*\\FFfpFA7$$!3iE5wfxCGFFfpFA7$$!3nv9,IqP2FFfpFA7$$!3OSs%zSC%)o#FfpF A7$$!3w6g<2z)em#FfpFA7$$!3O3^!\\`'zYEFfpFA7$$!3MN*z.t)eCEFfpFA7$$!3y$ \\2/1J\\g#FfpFA7$$!3=QY(3[jLe#FfpFA7$$!3c(3r[/EGc#FfpFA7$$!33)*)>\\Q(R TDFfpFA7$$!3QO5y&=><_#FfpFA7$$!335ud*e$\\+DFfpFA7$$!3?zpYghWyCFfpFA7$$ !3QaRh3QDfCFfpFA7$$!3iE5#GaD&QCFfpFA7$$!3\\lY*f@6rT#FfpFA7$$!3Cig*\\Zh hR#FfpFA7$$!3'=IkL_\"*eP#FfpFA7$$!3ix$4!)>&Q`BFfpFA7$$!3[>(oOEiJL#FfpF A7$$!3Gg6wk*p:J#FfpFA7$$!3%RG/F8/?H#FfpFA7$$!3#zj_-^81F#FfpFA7$$!3!y#e `,m[]AFfpFA7$$!3#>i@E=[%HAFfpFA7$$!3N:,m#Gz)3AFfpFA7$$!3a`oi]bM(=#FfpF A7$$!3;hSL2ggm@FfpFA7$$!3Ur&Q)GoRX@FfpFA7$$!3R2$\\IFjV7#FfpFA7$$!3as&R (>c.0@FfpFA7$$!3Mf?#z%R)G3#FfpFA7$$!3_ykdG72j?FfpFA7$$!37R;PPn%>/#FfpF A7$$!3/lU'Q!zs@?FfpFA7$$!30+++1+++?FfpFA7S7$$!3))*****f*******>FfpF+7$ $!3Y5x+UG?y>FfpF+7$$!3cox'[DP#f>FfpF+7$$!3Ql+eo%3z$>FfpF+7$$!3!p#=D,$Q k\">FfpF+7$$!3?06)yf_(=FfpF+7$$!3YJq.kgua=FfpF+7$ $!3X#3SvNJN$=FfpF+7$$!3OAf(oo%Q7=FfpF+7$$!3i/;%otK1z\"FfpF+7$$!3[W:T2O Zr<_\"FfpF+7$$!3]%Qz&*e$\\+ :FfpF+7$$!3;k2QghWy9FfpF+7$$!3kp4X3QDf9FfpF+7$$!35H^dUb_Q9FfpF+7$$!3F5 Jm:76<9FfpF+7$$!3E32eu9;'R\"FfpF+7$$!3'y'y'G_\"*eP\"FfpF+7$$!3_=HU(>&Q `8FfpF+7$$!3ko8+jA;L8FfpF+7$$!3ISu+k*p:J\"FfpF+7$$!3M+B(=8/?H\"FfpF+7$ $!3)=4N$4Nhq7FfpF+7$$!3;ux`+m[]7FfpF+7$$!3r9%R:=[%H7FfpF+7$$!3iKc\\\"G z)37FfpF+7$$!3qNiP\\bM(=\"FfpF+7$$!3C&[+g+1m;\"FfpF+7$$!3#*e,UFoRX6Ffp F+7$$!3TgnarKOC6FfpF+7$$!3'\\rf\"=c.06FfpF+7$$!3I&f`i%R)G3\"FfpF+7$$!3 kj(GoArI1\"FfpF+7$$!32E%RbtY>/\"FfpF+7$$!3\"p<^>!zs@5FfpF+7$$!36+++/++ +5FfpF+7S7$$!2%******z********FfpFA7$$!3E>_?R%G?y*!#=FA7$$!3)ff_()F^_nFA7$$!3Cd,ca1YZ&)F^_nFA7$$!3<_F^_nFA7$$!3A*e8e*e$\\+&F^_nFA7$$!3s'[X HghWy%F^_nFA7$$!3'y%)zG3QDf%F^_nFA7$$!3K6BHBaD&Q%F^_nFA7$$!3_^bJ`@6rTF ^_nFA7$$!3wSNlTZhhRF^_nFA7$$!3iQVrB_\"*ePF^_nFA7$$!3#>fk$o>&Q`$F^_nFA7 $$!3*oFA7$$ !3VX[53D72jF`gnFA7$$!3U-8sqLn%>%F`gnFA7$$!3EL)3Q+!zs@F`gnFA7$$!37+++++ ++?!#EFA7S7$$\"37+++++++?F]hnF+7$$\"3s/y%zg:(z@F`gnF+7$$\"3_O#G&HVFwSF `gnF+7$$\"3Lb+$o'H:4iF`gnF+7$$\"33))>C3(phN)F`gnF+7$$\"3]ShQ0#)H\\5F^_ nF+7$$\"3EMq#H13uC\"F^_nF+7$$\"3[U)RaMRDX\"F^_nF+7$$\"3+@zD6kok;F^_nF+ 7$$\"3a(QX(=J:w=F^_nF+7$$\"3yU'e*>En$4#F^_nF+7$$\"3+6c-:RE&G#F^_nF+7$$ \"3o)>'*\\L]4]#F^_nF+7$$\"355*HTQAvr#F^_nF+7$$\"3E\"3&H$oHi#HF^_nF+7$$ \"3nUOX0fv:JF^_nF+7$$\"3y\\)o\\\"47TLF^_nF+7$$\"3*[>0#RY.KNF^_nF+7$$\" 3))[NB'o7Tv$F^_nF+7$$\"3Uv0`(Q*o]RF^_nF+7$$\"3_RXe%=lj;%F^_nF+7$$\"31U IEY&RY2aF^_nF+7$$\"3n)o2ndW Zh&F^_nF+7$$\"3[[WoYy))GeF^_nF+7$$\"3BfkMe_QQgF^_nF+7$$\"3QhcGwZ3TiF^_ nF+7$$\"333ajJ![hY'F^_nF+7$$\"35B)fmPx$omF^_nF+7$$\"3%*)ziuO+V)oF^_nF+ 7$$\"3QJof*oe*zqF^_nF+7$$\"3cSX#e\"\\'QH(F^_nF+7$$\"3+(z-Y+M^\\(F^_nF+ 7$$\"3EFzU&>=bq(F^_nF+7$$\"3Z/&)o'>27\"zF^_nF+7$$\"3A@Qu=XaE\")F^_nF+7 $$\"3w24L`*RRL)F^_nF+7$$\"3GSD)*R<.Y&)F^_nF+7$$\"3_kyb*Hnjv)F^_nF+7$$ \"31D9?MQk\\*)F^_nF+7$$\"3!=p[Tbg6<*F^_nF+7$$\"3Y:&*=\\xGp$*F^_nF+7$$ \"3wpy#HmK0e*F^_nF+7$$\"3o;\">'**4s#y*F^_nF+7$$\"2%******z********FfpF +7S7$$\"36+++/+++5FfpFA7$$\"3a*G#*z:(z@5FfpFA7$$\"3WJA8XFwS5FfpFA7$$\" 3iM*>9`\"4i5FfpFA7$$\"35t\"[()phN3\"FfpFA7$$\"3z%*)=@#)H\\5\"FfpFA7$$ \"3USPz23uC6FfpFA7$$\"3aoH'f$RDX6FfpFA7$$\"3a<*fCkok;\"FfpFA7$$\"3kxS7 8`h(=\"FfpFA7$$\"3Q&ReJEn$47FfpFA7$$\"3`b%)e#RE&G7FfpFA7$$\"3')R#*\\M] 4]7FfpFA7$$\"39#)fKRAvr7FfpFA7$$\"3I;!f\"pHi#H\"FfpFA7$$\"34i!*H\"fv:J \"FfpFA7$$\"3eO/;#47TL\"FfpFA7$$\"3y0x]kM?`8FfpFA7$$\"3'*4<7p7Tv8FfpFA 7$$\"3u\"ys\"R*o]R\"FfpFA7$$\"3)y!>z=lj;9FfpFA7$$\"3N3w([&R'RQb@:FfpFA7$$\"3OI!\\:>Y2a\"FfpFA7$$\"3!4([UdWZh:FfpFA7$$\"3u*)o L%y))Ge\"FfpFA7$$\"3t\"H>a_QQg\"FfpFA7$$\"39K@8x%3Ti\"FfpFA7$$\"3[\"3x D![hY;FfpFA7$$\"3OJ')*ptPom\"FfpFA7$$\"3ofD*f.I%)o\"FfpFA7$$\"3m*pF\"o e*zq\"FfpFA7$$\"383\\m!\\'QHFfpFA7$$\"3NO7FfpFA7$$\"3#RdgWE`!e>FfpFA7$$\"35B)[! )4s#y>FfpFA7$$\"3))*****f*******>FfpFA7S7$$\"30+++1+++?FfpF+7$$\"3+,^! *frz@?FfpF+7$$\"3q!=ppui2/#FfpF+7$$\"3Mo:QPRDX@FfpF+7$ $\"3-VSzV'ok;#FfpF+7$$\"35;OP9`h(=#FfpF+7$$\"3ME4KksO4AFfpF+7$$\"3,]Vn $RE&GAFfpF+7$$\"3#)f))\\N]4]AFfpF+7$$\"3Qt*Q-C_lj;CFfpF+7$$ \"3W7*G^&R5!3:EgeCFfpF+7$$\"3ij*=U\"3GyCFfpF+7$$\"3#** eA/T1&*\\#FfpF+7$$\"3\"3-L&RQb@DFfpF+7$$\"3jXgQ\">Y2a#FfpF+7$$\"3Qt*yr Xu9c#FfpF+7$$\"3]M`+%y))Ge#FfpF+7$$\"3vPR+D&QQg#FfpF+7$$\"38)pNmZ3Ti#F fpF+7$$\"3QA1*>![hYEFfpF+7$$\"3_!GJjtPom#FfpF+7$$\"3sR)Q_.I%)o#FfpF+7$ $\"31;dHne*zq#FfpF+7$$\"34itu*['QHFFfpF+7$$\"3?sTY)R8&\\FFfpF+7$$\"33y $yt\"=bqFFfpF+7$$\"3k%))Rtr?6z#FfpF+7$$\"3YYJP\\Wl7GFfpF+7$$\"3$)Qfm#* RRLGFfpF+7$$\"3eG9;rJgaGFfpF+7$$\"3g#p]psOc(GFfpF+7$$\"3YF/E!Qk\\*GFfp F+7$$\"3nSz2_g6:TAvrKFfpFA7$$\"3 yK!=3(Hi#H$FfpFA7$$\"3>dk!Gfv:J$FfpFA7$$\"37Sv[$47TL$FfpFA7$$\"3%z2#ol M?`LFfpFA7$$\"3y>%=,F6aP$FfpFA7$$\"3YIA,S*o]R$FfpFA7$$\"3(f\")e%>lj;MF fpFA7$$\"3v;-QbRY2a$FfpFA7$$\"3SvI $pXu9c$FfpFA7$$\"3GzPn$y))Ge$FfpFA7$$\"3y$e)eC&QQg$FfpFA7$$\"3dk#RhZ3T i$FfpFA7$$\"31jTS,[hYOFfpFA7$$\"3WHRmNx$om$FfpFA7$$\"3_>^[M+V)o$FfpFA7 $$\"3pKPYme*zq$FfpFA7$$\"30;)H))['QHPFfpFA7$$\"3+>hY(R8&\\PFfpFA7$$\"3 '3<'H;=bqPFfpFA7$$\"3#>Svhr?6z$FfpFA7$$\"3jGD7[Wl7QFfpFA7$$\"3\"HOK8*R RLQFfpFA7$$\"3k:IupJgaQFfpFA7$$\"3kX\"[asOc(QFfpFA7$$\"3))p0oyV'\\*QFf pFA7$$\"3>w%4/0;r\"RFfpFA7$$\"3=1enp(Gp$RFfpFA7$$\"3%y9'zgK0eRFfpFA7$$ \"3;YEA%4s#yRFfpFA7$$\"3y*****>*******RFfpFA7S7$$\"3%*******4+++SFfpF+ 7$$\"3\\B2tjrz@SFfpF+7$$\"3LyIk]FwSSFfpF+7$$\"3!e$[nO:4iSFfpF+7$$\"3/L au.-%R DXTFfpF+7$$\"3_$Hikkok;%FfpF+7$$\"3]$psoJ:w=%FfpF+7$$\"3E))fkmsO4UFfpF +7$$\"3aQh%eRE&GUFfpF+7$$\"3q*4)\\P]4]UFfpF+7$$\"3'\\&\\1UAvrUFfpF+7$$ \"3!3aZ;(Hi#H%FfpF+7$$\"3>b,c$fv:J%FfpF+7$$\"3b\"4^T47TL%FfpF+7$$\"3!Q EpiY.KN%FfpF+7$$\"3#\\x;1F6aP%FfpF+7$$\"3sa>VS*o]R%FfpF+7$$\"3BpAz>lj; WFfpF+7$$\"31@:jbRY2a%FfpF+7$$\"3 WxrocWZhXFfpF+7$$\"3]CAM$y))Ge%FfpF+7$$\"3\")HK4,:27\"z%FfpF+7$$\"3C6>(oWaE\"[FfpF+7$$ \"3+(y)***)RRL[FfpF+7$$\"39.YKoJga[FfpF+7$$\"3m)fXRsOc([FfpF+7$$\"3I72 5xV'\\*[FfpF+7$$\"3;75u[g6<\\FfpF+7$$\"3)34GzwGp$\\FfpF+7$$\"3MMR'*eK0 e\\FfpF+7$$\"3#yb4B4s#y\\FfpF+7$$\"31+++!*******\\FfpF+7S7$$\"35+++7++ +]FfpFA7$$\"3tMNklrz@]FfpFA7$$\"3fF+[_FwS]FfpFA7$$\"3?qkUQ:4i]FfpFA7$$ \"3!f=6aqhN3&FfpFA7$$\"3>=+WG)H\\5&FfpFA7$$\"3#y)yz83uC^FfpFA7$$\"3#e! *Q;%RDX^FfpFA7$$\"3A>kzZ'ok;&FfpFA7$$\"3kKA7=`h(=&FfpFA7$$\"3x=&3yEn$4 _FfpFA7$$\"3ZL?$pRE&G_FfpFA7$$\"3k>x\\Q]4]_FfpFA7$$\"3UYz(HC_Y2a&FfpFA7$ $\"3#*z7WcWZhbFfpFA7$$\"3Ep1,$y))Ge&FfpFA7$$\"3&\\(yvB&QQg&FfpFA7$$\"3 c'RY^Z3Ti&FfpFA7$$\"3SW7B+[hYcFfpFA7$$\"3wF#HVtPom&FfpFA7$$\"3czw(H.I% )o&FfpFA7$$\"3^l(*zke*zq&FfpFA7$$\"3TCZ*p['QHdFfpFA7$$\"3s6+Z&R8&\\dFf pFA7$$\"3+c<89=bqdFfpFA7$$\"3cNk%Qr?6z&FfpFA7$$\"3(HH@cWaE\"eFfpFA7$$ \"336_m))RRLeFfpFA7$$\"3?!>1p;.Y&eFfpFA7$$\"37_IWAnjveFfpFA7$$\"3;b3_v V'\\*eFfpFA7$$\"3oZD2Zg6F-FCF0FFF3FGF5FHF8FKF;FR-%*AXESTICKSG6$%(DEFAULTGFJ-%+A XESLABELSG6%%\"xG%\"yG-%%FONTG6#F_`r-%%VIEWG6$;F(F<;$F`gnFB$\"#>FB" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 139 "The function f is discontinuou s at every integer, but is continuous everywhere else, that is, at all real numbers that are not integers. . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 68 "Br eaking down the checking of continuity of a function f at a point " } {XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG" }{TEXT -1 13 " into 3 steps" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "As sugges ted in the first section, a function f is " }{TEXT 261 10 "continuous " }{TEXT -1 4 " at " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG" }{TEXT -1 16 " provided that: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(f(x),x = a)=f(a)" "6#/-%&LimitG6$-%\"fG6#%\"xG/F*%\"aG-F(6#F, " }{TEXT -1 14 " ------- (i). " }}{PARA 0 "" 0 "" {TEXT -1 76 "Note th at there are three steps involved in checking that f is continuous a \+ " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG" }{TEXT -1 2 ". " }}{PARA 15 "" 0 "" {TEXT -1 18 " f is defined at " }{XPPEDIT 18 0 "x=a" "6#/%\"xG% \"aG" }{TEXT -1 11 ", that is, " }{TEXT 304 1 "a" }{TEXT -1 84 " belon gs to the domain of f. This ensures that the right-hand side of (i) \+ exists. " }}{PARA 15 "" 0 "" {XPPEDIT 18 0 "Limit(f(x),x = a)" "6#-%&L imitG6$-%\"fG6#%\"xG/F)%\"aG" }{TEXT -1 73 " exists (as a finite real \+ number). \nFor this to hold both the left limit " }{XPPEDIT 18 0 "Limi t(f(x),x = a^`-`)" "6#-%&LimitG6$-%\"fG6#%\"xG/F))%\"aG%\"-G" }{TEXT -1 17 " and right limit " }{XPPEDIT 18 0 "Limit(f(x),x = a^`+`)" "6#-% &LimitG6$-%\"fG6#%\"xG/F))%\"aG%\"+G" }{TEXT -1 124 " must exist and f urthermore they must have the same real number value. \nThis ensures t hat the left-hand side of (i) exists. " }}{PARA 15 "" 0 "" {TEXT -1 94 "Finally, the left and right sides of (i), as obtained in the first two points, must coincide. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 78 "Checking for continuity with refe rence to the definition using the three steps" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Exampl e 1 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "W e can say that the function " }{XPPEDIT 18 0 "f(x) = x+abs(x)/x;" "6#/ -%\"fG6#%\"xG,&F'\"\"\"*&-%$absG6#F'F)F'!\"\"F)" }{TEXT -1 21 " is dis continuous at " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 16 " si mply because " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 19 " \+ is not defined at " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "f := x -> x+abs(x)/x:\n'f(x)'=f(x);\np1 := plot(f(x) ,x=-3..3,discont=true):\np2 := plot([[0,-1],[0,1]],style=point,color=r ed,symbol=circle):\nplots[display]([p1,p2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&F'\"\"\"*&-%$absGF&F)F'!\"\"F)" }} {PARA 13 "" 1 "" {GLPLOT2D 315 318 318 {PLOTDATA 2 "6&-%'CURVESG6%7S7$ $!\"$\"\"!$!\"%F*7$$!3*GyIw`3Y$H!#<$!3*GyIw`3Y$RF07$$!3Ew&pex6x(GF0$!3 Ew&pex6x(QF07$$!3;\\Di;as8GF0$!3;\\Di;as8QF07$$!3?q8D9\\J\\FF0$!3?q8D9 \\J\\PF07$$!3yyXvV0@&o#F0$!3yyXvV0@&o$F07$$!3sWMP'exdi#F0$!3sWMP'exdi$ F07$$!3[Bl\\,#QUc#F0$!3[Bl\\,#QUc$F07$$!3p6Qi\"3%f+DF0$!3p6Qi\"3%f+NF0 7$$!3*=p]#pS:PCF0$!3*=p]#pS:PMF07$$!3m.iv=#)*=P#F0$!3m.iv=#)*=P$F07$$! 3ie67I3U9BF0$!3ie67I3U9LF07$$!3Sq0+/\\r\\AF0$!3Sq0+/\\r\\KF07$$!3S808* GVZ=#F0$!3S808*GVZ=$F07$$!3%ytb#*4J@7#F0$!3%ytb#*4J@7$F07$$!3Q`W\\KKFl ?F0$!3Q`W\\KKFlIF07$$!3ksY]HPm(*>F0$!3'Gn/&HPm(*HF07$$!3y?#>@h*QS>F0$! 3y?#>@h*QSHF07$$!3ynu(y>mP(=F0$!3+ou(y>mP(GF07$$!3t8/P(=$z9=F0$!3t8/P( =$z9GF07$$!3=>[7[/4][7[/4]FF07$$!3qV!)\\R\"y%)o\"F0$!3qV!)\\R \"y%)o#F07$$!3t:;+f@>C;F0$!3t:;+f@>CEF07$$!3%[o%*4cd^c\"F0$!3i%o%*4cd^ c#F07$$!3lQqur2[,:F0$!3lQqur2[,DF07$$!30BVv$[Q`V\"F0$!30BVv$[Q`V#F07$$ !3=x%>wUhxP\"F0$!3=x%>wUhxP#F07$$!3pY)='Hmd:8F0$!3#p%)='Hmd:BF07$$!3)H L(\\[OL^7F0$!3)HL(\\[OL^AF07$$!3IJI([U%[)=\"F0$!3IJI([U%[)=#F07$$!30^' p$pXnF6F0$!3$3lp$pXnF@F07$$!31*oHEfb,1\"F0$!3G*oHEfb,1#F07$$!3Ko-,!*y' [***!#=$!3%o-,!*y'[**>F07$$!3q-eI;*)4Z$*Ffu$!3F!eI;*)4Z$>F07$$!3Ua(H([ R7g()Ffu$!3WvH([R7g(=F07$$!3\"p=j(o_S=\")Ffu$!3p=j(o_S=\"=F07$$!3!p!3A ,!)f9vFfu$!3q!3A,!)f9v\"F07$$!3G5J[FaW$)oFfu$!3-6$[FaW$)o\"F07$$!3uWsY A%yjE'Ffu$!3[CnCUyjE;F07$$!37p#4]Xm.i&Ffu$!3#p#4]Xm.i:F07$$!3MRO+],=)* \\Ffu$!3%RO+],=)*\\\"F07$$!3#>>w()y/>O%Ffu$!3>>w()y/>O9F07$$!3;.#)y3\" )*3t$Ffu$!3K?)y3\")*3t8F07$$!3]iyp.&o5:$Ffu$!3D'yp.&o5:8F07$$!3Okp_U$= l[#Ffu$!3W'p_U$=l[7F07$$!3%)Hd@cn8#*=Ffu$!3)Hd@cn8#*=\"F07$$!3G&>LP,-% e7Ffu$!3`>LP,-%e7\"F07$$!3TrKYC+P=l!#>$!3sKYC+P=l5F07$$!3'************ ***f!#E$!30+++1+++5F07S7$$\"3'***************fFhz$\"30+++1+++5F07$$\"3 z1<#p$o9RlFbz$\"3;<#p$o9Rl5F07$$\"3XNUI,B)GA\"Ffu$\"3cB/8I#)GA6F07$$\" 353Xx$*eui=Ffu$\"3)3Xx$*eui=\"F07$$\"3^)H'[<4&o]#Ffu$\"3')H'[<4&o]7F07 $$\"3O5UXAY*y9$Ffu$\"3/@aCi%*y98F07$$\"39^bE'>CAu$Ffu$\"37bli>CAu8F07$ $\"3YjZ.X!=wN%Ffu$\"3NwM]/=wN9F07$$\"3O\")=wV#fS*\\Ffu$\"39)=wV#fS*\\ \"F07$$\"3i\"3$\\n$f%GcFfu$\"3;3$\\n$f%Gc\"F07$$\"3/lzVsy,\"G'Ffu$\"3S 'zVsy,\"G;F07$$\"3_;%)ye5&G](Ffu $\"3))H%**>5&G]F07$$\"3UF`\\wiL- 5F0$\"3kF`\\wiL-?F07$$\"3Ez2)QR5'f5F0$\"3Ez2)QR5'f?F07$$\"3FKD73QBE6F0 $\"3\\KD73QBE@F07$$\"3K'eH'=o?&=\"F0$\"3K'eH'=o?&=#F07$$\"3)3=vyb4*\\7 F0$\"3)3=vyb4*\\AF07$$\"3Nc>]m=_68F0$\"3Nc>]m=_6BF07$$\"3K%Q)*p%y!eP\" F0$\"3K%Q)*p%y!eP#F07$$\"3@:`+XC%[V\"F0$\"3*\\J0]WU[V#F07$$\"3ShHDM#>& )\\\"F0$\"3ShHDM#>&)\\#F07$$\"3+xcCA:mk:F0$\"3+xcCA:mkDF07$$\"3)G_!Qy& QAi\"F0$\"3)G_!Qy&QAi#F07$$\"3O`6QwLU%o\"F0$\"3e`6QwLU%o#F07$$\"33nE]d jm[F0$\"3@6.P8W%)RHF07$$\"3At*)*p@8 0+#F0$\"3At*)*p@80+$F07$$\"3y>%pV6!Hl?F0$\"3y>%pV6!HlIF07$$\"3hCq76w)R 7#F0$\"3hCq76w)R7$F07$$\"3O\"oB\"z%f\")=#F0$\"3O\"oB\"z%f\")=$F07$$\"3 O>z(e?S&[AF0$\"3O>z(e?S&[KF07$$\"3.*o^KYb;J#F0$\"3.*o^KYb;J$F07$$\"3ev Kvj@OtBF0$\"3evKvj@OtLF07$$\"39t!*\\gL'zV#F0$\"39t!*\\gL'zV$F07$$\"37O '**4*>=+DF0$\"37O'**4*>=+NF07$$\"3'3QAr_4Qc#F0$\"3'3QAr_4Qc$F07$$\"3uz 67&>5pi#F0$\"3uz67&>5pi$F07$$\"3!Q@Ic:$*[o#F0$\"3!Q@Ic:$*[o$F07$$\"3i. tur\"[8v#F0$\"3i.tur\"[8v$F07$$\"32F%y.L'y5GF0$\"32F%y.L'y5QF07$$\"3_! oEY!)fT(GF0$\"3_!oEY!)fT(QF07$$\"3Mn`v0j\"[$HF0$\"3Mn`v0j\"[$RF07$$\" \"$F*$\"\"%F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$F*F*Fhjl-F$6&7$7$Fhjl$! \"\"F*7$Fhjl$\"\"\"F*Fajl-%'SYMBOLG6#%'CIRCLEG-%&STYLEG6#%&POINTG-%+AX ESLABELSG6%Q\"x6\"Q!F^\\m-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F(F]jlFc\\m" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Cur ve 2" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "Now consider the follo wing modification or extension of " }{XPPEDIT 18 0 "f(x) = x+abs(x)/x " "6#/-%\"fG6#%\"xG,&F'\"\"\"*&-%$absG6#F'F)F'!\"\"F)" }{TEXT -1 15 " \+ to a function " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 59 " defined for all real numbers, in which we give a value at " } {XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "g(x);" "6#-%\"gG6#%\"xG" }{TEXT -1 63 " be the functon described by the following piecewise formula: " } }{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g(x) = PIECEWISE([x+ 1, 0 <= x],[x-1, x < 0]);" "6#/-%\"gG6#%\"xG-%*PIECEWISEG6$7$,&F'\"\" \"F-F-1\"\"!F'7$,&F'F-F-!\"\"2F'F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 267 "g := x -> piecewise(x<0,x-1,x>=0,x+1):\n'g(x)'=g(x);\np1 := plot(g(x),x=-3..3,d iscont=true):\np2 := plot([[0,-1]],style=point,color=red,symbol=circle ):\np3 := plot([[[0,1]]$3],style=point,symbol=[circle,diamond,cross],c olor=red,symbol=circle):\nplots[display]([p1,p2,p3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG-%*PIECEWISEG6$7$,&F'\"\"\"F-!\"\"2F '\"\"!7$,&F'F-F-F-1F0F'" }}{PARA 13 "" 1 "" {GLPLOT2D 296 316 316 {PLOTDATA 2 "6*-%'CURVESG6%7S7$$!\"$\"\"!$!\"%F*7$$!3*GyIw`3Y$H!#<$!3* GyIw`3Y$RF07$$!3Ew&pex6x(GF0$!3Ew&pex6x(QF07$$!3;\\Di;as8GF0$!3;\\Di;a s8QF07$$!3?q8D9\\J\\FF0$!3?q8D9\\J\\PF07$$!3yyXvV0@&o#F0$!3yyXvV0@&o$F 07$$!3sWMP'exdi#F0$!3sWMP'exdi$F07$$!3[Bl\\,#QUc#F0$!3[Bl\\,#QUc$F07$$ !3p6Qi\"3%f+DF0$!3p6Qi\"3%f+NF07$$!3*=p]#pS:PCF0$!3*=p]#pS:PMF07$$!3m. iv=#)*=P#F0$!3m.iv=#)*=P$F07$$!3ie67I3U9BF0$!3ie67I3U9LF07$$!3Sq0+/\\r \\AF0$!3Sq0+/\\r\\KF07$$!3S808*GVZ=#F0$!3S808*GVZ=$F07$$!3%ytb#*4J@7#F 0$!3%ytb#*4J@7$F07$$!3Q`W\\KKFl?F0$!3Q`W\\KKFlIF07$$!3ksY]HPm(*>F0$!3' Gn/&HPm(*HF07$$!3y?#>@h*QS>F0$!3y?#>@h*QSHF07$$!3ynu(y>mP(=F0$!3+ou(y> mP(GF07$$!3t8/P(=$z9=F0$!3t8/P(=$z9GF07$$!3=>[7[/4][7[/4]FF07$ $!3qV!)\\R\"y%)o\"F0$!3qV!)\\R\"y%)o#F07$$!3t:;+f@>C;F0$!3t:;+f@>CEF07 $$!3%[o%*4cd^c\"F0$!3i%o%*4cd^c#F07$$!3lQqur2[,:F0$!3lQqur2[,DF07$$!30 BVv$[Q`V\"F0$!30BVv$[Q`V#F07$$!3=x%>wUhxP\"F0$!3=x%>wUhxP#F07$$!3pY)=' Hmd:8F0$!3#p%)='Hmd:BF07$$!3)HL(\\[OL^7F0$!3)HL(\\[OL^AF07$$!3IJI([U%[ )=\"F0$!3IJI([U%[)=#F07$$!30^'p$pXnF6F0$!3$3lp$pXnF@F07$$!31*oHEfb,1\" F0$!3G*oHEfb,1#F07$$!3Ko-,!*y'[***!#=$!3%o-,!*y'[**>F07$$!3q-eI;*)4Z$* Ffu$!3F!eI;*)4Z$>F07$$!3Ua(H([R7g()Ffu$!3WvH([R7g(=F07$$!3\"p=j(o_S=\" )Ffu$!3p=j(o_S=\"=F07$$!3!p!3A,!)f9vFfu$!3q!3A,!)f9v\"F07$$!3G5J[FaW$) oFfu$!3-6$[FaW$)o\"F07$$!3uWsYA%yjE'Ffu$!3[CnCUyjE;F07$$!37p#4]Xm.i&Ff u$!3#p#4]Xm.i:F07$$!3MRO+],=)*\\Ffu$!3%RO+],=)*\\\"F07$$!3#>>w()y/>O%F fu$!3>>w()y/>O9F07$$!3;.#)y3\")*3t$Ffu$!3K?)y3\")*3t8F07$$!3]iyp.&o5:$ Ffu$!3D'yp.&o5:8F07$$!3Okp_U$=l[#Ffu$!3W'p_U$=l[7F07$$!3%)Hd@cn8#*=Ffu $!3)Hd@cn8#*=\"F07$$!3G&>LP,-%e7Ffu$!3`>LP,-%e7\"F07$$!3TrKYC+P=l!#>$! 3sKYC+P=l5F07$$!3'***************f!#E$!30+++1+++5F07S7$$\"3'********** *****fFhz$\"30+++1+++5F07$$\"3z1<#p$o9RlFbz$\"3;<#p$o9Rl5F07$$\"3XNUI, B)GA\"Ffu$\"3cB/8I#)GA6F07$$\"353Xx$*eui=Ffu$\"3)3Xx$*eui=\"F07$$\"3^) H'[<4&o]#Ffu$\"3')H'[<4&o]7F07$$\"3O5UXAY*y9$Ffu$\"3/@aCi%*y98F07$$\"3 9^bE'>CAu$Ffu$\"37bli>CAu8F07$$\"3YjZ.X!=wN%Ffu$\"3NwM]/=wN9F07$$\"3O \")=wV#fS*\\Ffu$\"39)=wV#fS*\\\"F07$$\"3i\"3$\\n$f%GcFfu$\"3;3$\\n$f%G c\"F07$$\"3/lzVsy,\"G'Ffu$\"3S'zVsy,\"G;F07$$\"3_;%)ye5&G](Ffu$\"3))H%**>5&G]F07$$\"3UF`\\wiL-5F0$\"3kF`\\wiL-?F07$$\"3Ez2)QR5'f5F0$ \"3Ez2)QR5'f?F07$$\"3FKD73QBE6F0$\"3\\KD73QBE@F07$$\"3K'eH'=o?&=\"F0$ \"3K'eH'=o?&=#F07$$\"3)3=vyb4*\\7F0$\"3)3=vyb4*\\AF07$$\"3Nc>]m=_68F0$ \"3Nc>]m=_6BF07$$\"3K%Q)*p%y!eP\"F0$\"3K%Q)*p%y!eP#F07$$\"3@:`+XC%[V\" F0$\"3*\\J0]WU[V#F07$$\"3ShHDM#>&)\\\"F0$\"3ShHDM#>&)\\#F07$$\"3+xcCA: mk:F0$\"3+xcCA:mkDF07$$\"3)G_!Qy&QAi\"F0$\"3)G_!Qy&QAi#F07$$\"3O`6QwLU %o\"F0$\"3e`6QwLU%o#F07$$\"33nE]djm[F0$\"3@6.P8W%)RHF07$$\"3At*)*p@80+#F0$\"3At*)*p@80+$F07$$\"3y>%pV6!H l?F0$\"3y>%pV6!HlIF07$$\"3hCq76w)R7#F0$\"3hCq76w)R7$F07$$\"3O\"oB\"z%f \")=#F0$\"3O\"oB\"z%f\")=$F07$$\"3O>z(e?S&[AF0$\"3O>z(e?S&[KF07$$\"3.* o^KYb;J#F0$\"3.*o^KYb;J$F07$$\"3evKvj@OtBF0$\"3evKvj@OtLF07$$\"39t!*\\ gL'zV#F0$\"39t!*\\gL'zV$F07$$\"37O'**4*>=+DF0$\"37O'**4*>=+NF07$$\"3'3 QAr_4Qc#F0$\"3'3QAr_4Qc$F07$$\"3uz67&>5pi#F0$\"3uz67&>5pi$F07$$\"3!Q@I c:$*[o#F0$\"3!Q@Ic:$*[o$F07$$\"3i.tur\"[8v#F0$\"3i.tur\"[8v$F07$$\"32F %y.L'y5GF0$\"32F%y.L'y5QF07$$\"3_!oEY!)fT(GF0$\"3_!oEY!)fT(QF07$$\"3Mn `v0j\"[$HF0$\"3Mn`v0j\"[$RF07$$\"\"$F*$\"\"%F*-%'COLOURG6&%$RGBG$\"*++ ++\"!\")$F*F*Fhjl-%'POINTSG6$7$Fhjl$\"\"\"F*Fajl-F$6&7#7$Fhjl$!\"\"F*F ajl-%'SYMBOLG6#%'CIRCLEG-%&STYLEG6#%&POINTG-F$6&7#F\\[mFe[mFajlFi[m-F$ 6&F_\\m-Ff[m6#%(DIAMONDGFajlFi[m-F$6&F_\\m-Ff[m6#%&CROSSGFajlFi[m-%+AX ESLABELSG6%Q\"x6\"Q!F^]m-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F(F]jlFc]m" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 39.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "It is intuitively clear that g is discontinuous at " } {XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 10 ", because " } {XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 27 " still \"jumps\" \+ in value as " }{TEXT 305 1 "x" }{TEXT -1 22 " increases through 0. " } }{PARA 0 "" 0 "" {TEXT -1 31 "We can check for continuity at " } {XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 68 " with reference to t he 3 step analysis to see whether the condition " }{XPPEDIT 18 0 "Limi t(g(x),x = 0)=g(0)" "6#/-%&LimitG6$-%\"gG6#%\"xG/F*\"\"!-F(6#F," } {TEXT -1 8 " holds. " }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g(0)" "6#-%\"gG6#\"\"!" }{TEXT -1 12 " exists and " }{XPPEDIT 18 0 "g(0)=1" "6#/-%\"gG6#\"\"!\"\"\"" }{TEXT -1 2 ". " }}{PARA 15 "" 0 "" {XPPEDIT 18 0 "Limit(g(x),x=0)" "6#-%&LimitG6$-%\"gG6#%\"xG/F)\"\"!" } {TEXT -1 24 " does not exist because " }{XPPEDIT 18 0 "Limit(g(x),x = \+ 0^`-`) = -1;" "6#/-%&LimitG6$-%\"gG6#%\"xG/F*)\"\"!%\"-G,$\"\"\"!\"\" " }{TEXT -1 7 " while " }{XPPEDIT 18 0 "Limit(g(x),x = 0^`+`) = 1;" "6 #/-%&LimitG6$-%\"gG6#%\"xG/F*)\"\"!%\"+G\"\"\"" }{TEXT -1 2 ". " }} {PARA 15 "" 0 "" {TEXT -1 40 "not applicable because of the 2nd step. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "There fore, " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 22 " is not \+ continuous at " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2 " }}{PARA 0 "" 0 "" {TEXT -1 69 "As for the previous example, we can say that th e function f given by " }{XPPEDIT 18 0 "f(x) = x/(x-1);" "6#/-%\"fG6#% \"xG*&F'\"\"\",&F'F)F)!\"\"F+" }{TEXT -1 21 " is discontinuous at " } {XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 16 " simply because \+ " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 19 " is not define d at " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "f := x -> x/(x-1):\n'f(x)'=f(x);\np1 := plot(f(x),x=-3..5,y=-3..5 ,discont=true,thickness=2):\np2 := plot([[[-3,1],[5,1]],[[1,-3],[1,5]] ],color=black,linestyle=3):\nplots[display]([p1,p2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*&F'\"\"\",&F'F)F)!\"\"F+" }}{PARA 13 "" 1 "" {GLPLOT2D 339 306 306 {PLOTDATA 2 "6'-%'CURVESG6&7gn7$$!\"$ \"\"!$\"3++++++++v!#=7$$!31;TY$Q6G\"H!#<$\"3B&\\$\\dGHWuF-7$$!3\"HC6W. \\p$GF1$\"3!p*[k>Ew$R(F-7$$!3m#)eq))Qj^FF1$\"3mxIt.Z\\MtF-7$$!3s.Z$)=K vlEF1$\"3)3H=%zw/ssF-7$$!3QXizC2G!e#F1$\"3=Wz/HH#p?(F-7$$!3+yZ:vqF-7$$!3KX@$=WDTL#F1$\" 3$=\\\"GGFr+qF-7$$!3'=pD'e(Q&\\AF1$\"3(4PqG=SE#pF-7$$!3?qG#z&4`i@F1$\" 3nT')H'**>F1$ \"3mfG9[UCmmF-7$$!3`80j^5*H\">F1$\"3=)palR-rc'F-7$$!3zPdvJ\"3&H=F1$\"3 OEReOo\"eY'F-7$$!3%oy-Fk(p`&ojF-7$$!39R8nQ;bj;F1$\"3'f;q )oVhXiF-7$$!3Q()ey[h=(e\"F1$\"3o<&p+(ozMhF-7$$!3knCvH\\N)\\\"F1$\"3at& fW:mt*fF-7$$!3]!3P!\\Us>9F1$\"3aWzL(o(HneF-7$$!3:>)***HRXL8F1$\"3,F$>p ?2Xr&F-7$$!3wVI7&=/8D\"F1$\"3af.KJ-8ebF-7$$!3A#G=Wa*el6F1$\"3I&[+=y>BQ &F-7$$!3S^j.Zn(o3\"F1$\"364+1j,:3_F-7$$!3xr.LhV(>+\"F1$\"3(=-.b.J\\+&F -7$$!3SiliRk%y8*F-$\"3q\\rvx;vuZF-7$$!3QtZWdB:q$)F-$\"3e)))o+,(QcXF-7$ $!3!3!=-<<-TvF-$\"3=GWo1y2*H%F-7$$!3'yKt\\j[Wo'F-$\"38)=%)fo%R1SF-7$$! 3!=JIi)*ek%eF-$\"35T)fq!>W*o$F-7$$!3E6lW74mN]F-$\"3Ed<5P]9\\LF-7$$!3D* )oz))ySNTF-$\"32Oxj>ncDHF-7$$!3pNpn10\\ELF-$\"3'fG#fL*[h\\#F-7$$!3+/eI T&)ziCF-$\"3m!Qfn**>h(>F-7$$!3?)383Dl,o\"F-$\"3e*o*R%Hx%Q9F-7$$!3)H*=j PMSX#)!#>$\"3&>ExlJCth(Ffv7$$!3wsvuj')RY>!#?$\"365L`bvhU>F\\w7$$\"3K8* owyF2A)Ffv$!3)=(4N/O1d*)Ffv7$$\"3;)3meyG[k\"F-$!3gDpPVcjo>F-7$$\"3TI2C w!yh]#F-$!3,2/#[jDVM$F-7$$\"3!4.jm\")fdL$F-$!3V_]Nv:Y0]F-7$$\"3!yZ!Rlp 7%=%F-$!3K18r)3BV>(F-7$$\"3BK^z0#pa-&F-$!3N\")*[C%)R-,\"F17$$\"3WR@IY` d)z&F-$!3e!H'*z-X,Q\"F17$$\"3AsjIGAk%o'F-$!3o.v6@dE;?F17$$\"3yR4X55:xu F-$!3)ehB%>FxjHF17$$\"3(\\!o^n18A$)F-$!3IQr`s'R*f\\F17$$\"3.R-\">M2ls) F-$!3#>.tQh?C&oF17$$\"33tOI;S)38*F-$!3%oU\\'=Wf]5!#;7$$\"3oaxA(*H;[$*F -$!3&eVHi&e7M9F[[l7$$\"3GO=:y>Wl&*F-$!3JhZVy')=,AF[[l7$$\"3kxQho93u'*F -$!3EVQ.I9DoHF[[l7$$\"3)y\"f2f4s#y*F-$!3g%G89swB]%F[[l7$$\"3bQpI/2/P)* F-$!38kyOI<]OgF[[l7$$\"37ez`\\/O\"*)*F-$!3@BE+,4v/\"*F[[l7$$\"3\"zY`@K ?&=**F-$!3kwQ>%*)*H<7!#:7$$\"3!)y*oZ>!oX**F-$!3r2=Zj\"\\4$=F_]l7$$\"3q *[%Qn+%G(**F-$!3Ogb.gU*=n$F_]l7$$\"2%******R********F1$!3KiT7Ummm;!\"* 7gn7$$\"3%*******4+++5F1$\"33ru2;+++5F_^l7$$\"3!=-kUlCF+\"F1$\"3!)o?@G />!o$F_]l7$$\"3VV!G&)H\\a+\"F1$\"3W8no\"e)4X=F_]l7$$\"32l?zUR<35F1$\"3 #>!4\"G!)*RL7F_]l7$$\"3$p3cqe)*3,\"F1$\"3]^#3uK,bF*F[[l7$$\"3?ITevyM;5 F1$\"3xpW#ev-q@'F[[l7$$\"3pt@6krz@5F1$\"3iS2XoFv(o%F[[l7$$\"3Yg#o6u&pK 5F1$\"3G$ebLJ-&eJF[[l7$$\"3XZVA=VfV5F1$\"3KuBT5p(QR#F[[l7$$\"3?@lLs9Rl 5F1$\"3hI(Q0R^#H;F[[l7$$\"3'\\p[ki)=(3\"F1$\"3c$)Hv'eQpC\"F[[l7$$\"3K+ s$4!)>^7\"F1$\"3;_EM\"3SB**)F17$$\"3!fqDa(40j6F1$\"3;$G?'>80LrF17$$\"3 I^d/@hO[7F1$\"3**Qm3RSJE]F17$$\"3!)\\5$3zYUL\"F1$\"3!H$e()\\8!=*RF17$$ \"3>NSy%G>(>9F1$\"3I'Hx8![a#Q$F17$$\"3ie#>zAj*)\\\"F1$\"3&[bh`qbT+$F17 $$\"3cgu/Td,\"e\"F1$\"3Ux,UlP7@FF17$$\"39!)>]nX(em\"F1$\"3+lc=VVy,DF17 $$\"3%o%Qi]7Y]#F1 7$$\"3k-9Qpb59>F1$\"3qP)[yalR4#F17$$\"3v\\!*\\P,Q+?F1$\"3kd0&p+?'**>F1 7$$\"3qxCGd*3q3#F1$\"3V**pp?c&*>>F17$$\"3AqP2x=\\q@F1$\"3ol;$GoTV&=F17 $$\"3]540mBIYAF1$\"3:>\\\"o]tB!=F17$$\"3s7A**p$[kL#F1$\"39!=#)f(=D[0Oy*eC$F1$\"3g3eYGiDX9F17$$\"3?76Y%)Qr0S\"F1 7$$\"3'>&Q`=#fke$F1$\"3'o,2a%*GmQ\"F17$$\"3eb\\mc4NnOF1$\"3sucS'z.\\P \"F17$$\"3i*p:J:?Pv$F1$\"3GWm72^9j8F17$$\"3-3n3#[$)>$QF1$\"3'QWxTS4JN \"F17$$\"3DNhqsfaQ$3O0)*RF1$\"3?QsZR(\\NL \"F17$$\"3q\"[%z%G2A3%F1$\"3>B%RC\"GWC8F17$$\"3IE@U&)G[kTF1$\"3bUk;(R2 gJ\"F17$$\"3)eXtV\"yh]UF1$\"3tqv]FQj28F17$$\"3RFFL))fdLVF1$\"3)ztRE;y* *H\"F17$$\"3]N17.FT=WF1$\"3'zQ*)>VLDH\"F17$$\"3\"p'p2Fpa-XF1$\"3UCj[Cl ]&G\"F17$$\"3-d.0Tv&)zXF1$\"3ofZuC2Mz7F17$$\"3'HH28A$[F1$\"3APr]Ce%4E \"F17$$\"3tysr2%)38\\F1$\"3qjuI@Ebb7F17$$\"\"&F*$\"3+++++++]7F1-%'COLO URG6&%$RGBG$\"*++++\"!\")$F*F*F_am-%*THICKNESSG6#\"\"#-F$6%7$7$F($\"\" \"F*7$Fd`mFham-Fi`m6&F[amF*F*F*-%*LINESTYLEG6#\"\"$-F$6%7$7$FhamF(7$Fh amFd`mF[bmF]bm-%+AXESLABELSG6%Q\"x6\"Q\"yFjbm-%%FONTG6#%(DEFAULTG-%%VI EWG6$;F(Fd`mFccm" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "Now consider the following modification or extension of \+ " }{XPPEDIT 18 0 "f(x) = x/(x-1);" "6#/-%\"fG6#%\"xG*&F'\"\"\",&F'F)F) !\"\"F+" }{TEXT -1 15 " to a function " }{XPPEDIT 18 0 "g(x)" "6#-%\"g G6#%\"xG" }{TEXT -1 59 " defined for all real numbers, in which we giv e a value at " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "g(x);" "6#-%\"gG 6#%\"xG" }{TEXT -1 63 " be the functon described by the following pie cewise formula: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "g(x) = PIECEWISE([ x/(x-1), x <> 1],[1, x = 1]);" "6#/-%\"gG6#%\"xG-%*PIECEWISEG6$7$*&F' \"\"\",&F'F-F-!\"\"F/0F'F-7$F-/F'F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 264 "f := x -> x/(x-1):\n'f(x)'=f(x);\np1 := plot(f(x),x=-3..5,y=-3..5,discont=true, thickness=2):\np2 := plot([[[-3,1],[5,1]],[[1,-3],[1,5]]],color=black, linestyle=3):\np3 := plot([[[1,1]]$3],color=red,style=point,symbol=[ci rcle,diamond,cross]):\nplots[display]([p1,p2,p3]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"fG6#%\"xG*&F'\"\"\",&F'F)F)!\"\"F+" }}{PARA 13 " " 1 "" {GLPLOT2D 344 290 290 {PLOTDATA 2 "6*-%'CURVESG6&7gn7$$!\"$\"\" !$\"3++++++++v!#=7$$!31;TY$Q6G\"H!#<$\"3B&\\$\\dGHWuF-7$$!3\"HC6W.\\p$ GF1$\"3!p*[k>Ew$R(F-7$$!3m#)eq))Qj^FF1$\"3mxIt.Z\\MtF-7$$!3s.Z$)=KvlEF 1$\"3)3H=%zw/ssF-7$$!3QXizC2G!e#F1$\"3=Wz/HH#p?(F-7$$!3+yZ:vqF-7$$!3KX@$=WDTL#F1$\"3$= \\\"GGFr+qF-7$$!3'=pD'e(Q&\\AF1$\"3(4PqG=SE#pF-7$$!3?qG#z&4`i@F1$\"3nT ')H'**>F1$\"3m fG9[UCmmF-7$$!3`80j^5*H\">F1$\"3=)palR-rc'F-7$$!3zPdvJ\"3&H=F1$\"3OERe Oo\"eY'F-7$$!3%oy-Fk(p`&ojF-7$$!39R8nQ;bj;F1$\"3'f;q)oVh XiF-7$$!3Q()ey[h=(e\"F1$\"3o<&p+(ozMhF-7$$!3knCvH\\N)\\\"F1$\"3at&fW:m t*fF-7$$!3]!3P!\\Us>9F1$\"3aWzL(o(HneF-7$$!3:>)***HRXL8F1$\"3,F$>p?2Xr &F-7$$!3wVI7&=/8D\"F1$\"3af.KJ-8ebF-7$$!3A#G=Wa*el6F1$\"3I&[+=y>BQ&F-7 $$!3S^j.Zn(o3\"F1$\"364+1j,:3_F-7$$!3xr.LhV(>+\"F1$\"3(=-.b.J\\+&F-7$$ !3SiliRk%y8*F-$\"3q\\rvx;vuZF-7$$!3QtZWdB:q$)F-$\"3e)))o+,(QcXF-7$$!3! 3!=-<<-TvF-$\"3=GWo1y2*H%F-7$$!3'yKt\\j[Wo'F-$\"38)=%)fo%R1SF-7$$!3!=J Ii)*ek%eF-$\"35T)fq!>W*o$F-7$$!3E6lW74mN]F-$\"3Ed<5P]9\\LF-7$$!3D*)oz) )ySNTF-$\"32Oxj>ncDHF-7$$!3pNpn10\\ELF-$\"3'fG#fL*[h\\#F-7$$!3+/eIT&)z iCF-$\"3m!Qfn**>h(>F-7$$!3?)383Dl,o\"F-$\"3e*o*R%Hx%Q9F-7$$!3)H*=jPMSX #)!#>$\"3&>ExlJCth(Ffv7$$!3wsvuj')RY>!#?$\"365L`bvhU>F\\w7$$\"3K8*owyF 2A)Ffv$!3)=(4N/O1d*)Ffv7$$\"3;)3meyG[k\"F-$!3gDpPVcjo>F-7$$\"3TI2Cw!yh ]#F-$!3,2/#[jDVM$F-7$$\"3!4.jm\")fdL$F-$!3V_]Nv:Y0]F-7$$\"3!yZ!Rlp7%=% F-$!3K18r)3BV>(F-7$$\"3BK^z0#pa-&F-$!3N\")*[C%)R-,\"F17$$\"3WR@IY`d)z& F-$!3e!H'*z-X,Q\"F17$$\"3AsjIGAk%o'F-$!3o.v6@dE;?F17$$\"3yR4X55:xuF-$! 3)ehB%>FxjHF17$$\"3(\\!o^n18A$)F-$!3IQr`s'R*f\\F17$$\"3.R-\">M2ls)F-$! 3#>.tQh?C&oF17$$\"33tOI;S)38*F-$!3%oU\\'=Wf]5!#;7$$\"3oaxA(*H;[$*F-$!3 &eVHi&e7M9F[[l7$$\"3GO=:y>Wl&*F-$!3JhZVy')=,AF[[l7$$\"3kxQho93u'*F-$!3 EVQ.I9DoHF[[l7$$\"3)y\"f2f4s#y*F-$!3g%G89swB]%F[[l7$$\"3bQpI/2/P)*F-$! 38kyOI<]OgF[[l7$$\"37ez`\\/O\"*)*F-$!3@BE+,4v/\"*F[[l7$$\"3\"zY`@K?&=* *F-$!3kwQ>%*)*H<7!#:7$$\"3!)y*oZ>!oX**F-$!3r2=Zj\"\\4$=F_]l7$$\"3q*[%Q n+%G(**F-$!3Ogb.gU*=n$F_]l7$$\"2%******R********F1$!3KiT7Ummm;!\"*7gn7 $$\"3%*******4+++5F1$\"33ru2;+++5F_^l7$$\"3!=-kUlCF+\"F1$\"3!)o?@G/>!o $F_]l7$$\"3VV!G&)H\\a+\"F1$\"3W8no\"e)4X=F_]l7$$\"32l?zUR<35F1$\"3#>!4 \"G!)*RL7F_]l7$$\"3$p3cqe)*3,\"F1$\"3]^#3uK,bF*F[[l7$$\"3?ITevyM;5F1$ \"3xpW#ev-q@'F[[l7$$\"3pt@6krz@5F1$\"3iS2XoFv(o%F[[l7$$\"3Yg#o6u&pK5F1 $\"3G$ebLJ-&eJF[[l7$$\"3XZVA=VfV5F1$\"3KuBT5p(QR#F[[l7$$\"3?@lLs9Rl5F1 $\"3hI(Q0R^#H;F[[l7$$\"3'\\p[ki)=(3\"F1$\"3c$)Hv'eQpC\"F[[l7$$\"3K+s$4 !)>^7\"F1$\"3;_EM\"3SB**)F17$$\"3!fqDa(40j6F1$\"3;$G?'>80LrF17$$\"3I^d /@hO[7F1$\"3**Qm3RSJE]F17$$\"3!)\\5$3zYUL\"F1$\"3!H$e()\\8!=*RF17$$\"3 >NSy%G>(>9F1$\"3I'Hx8![a#Q$F17$$\"3ie#>zAj*)\\\"F1$\"3&[bh`qbT+$F17$$ \"3cgu/Td,\"e\"F1$\"3Ux,UlP7@FF17$$\"39!)>]nX(em\"F1$\"3+lc=VVy,DF17$$ \"3%o%Qi]7Y]#F17$ $\"3k-9Qpb59>F1$\"3qP)[yalR4#F17$$\"3v\\!*\\P,Q+?F1$\"3kd0&p+?'**>F17$ $\"3qxCGd*3q3#F1$\"3V**pp?c&*>>F17$$\"3AqP2x=\\q@F1$\"3ol;$GoTV&=F17$$ \"3]540mBIYAF1$\"3:>\\\"o]tB!=F17$$\"3s7A**p$[kL#F1$\"39!=#)f(=D[0Oy*eC$F1$\"3g3eYGiDX9F17$$\"3?76Y%)Qr0S\"F17$$ \"3'>&Q`=#fke$F1$\"3'o,2a%*GmQ\"F17$$\"3eb\\mc4NnOF1$\"3sucS'z.\\P\"F1 7$$\"3i*p:J:?Pv$F1$\"3GWm72^9j8F17$$\"3-3n3#[$)>$QF1$\"3'QWxTS4JN\"F17 $$\"3DNhqsfaQ$3O0)*RF1$\"3?QsZR(\\NL\"F17$ $\"3q\"[%z%G2A3%F1$\"3>B%RC\"GWC8F17$$\"3IE@U&)G[kTF1$\"3bUk;(R2gJ\"F1 7$$\"3)eXtV\"yh]UF1$\"3tqv]FQj28F17$$\"3RFFL))fdLVF1$\"3)ztRE;y**H\"F1 7$$\"3]N17.FT=WF1$\"3'zQ*)>VLDH\"F17$$\"3\"p'p2Fpa-XF1$\"3UCj[Cl]&G\"F 17$$\"3-d.0Tv&)zXF1$\"3ofZuC2Mz7F17$$\"3'HH28A$[F1$\"3APr]Ce%4E\"F17$$ \"3tysr2%)38\\F1$\"3qjuI@Ebb7F17$$\"\"&F*$\"3+++++++]7F1-%'COLOURG6&%$ RGBG$\"*++++\"!\")$F*F*F_am-%*THICKNESSG6#\"\"#-F$6%7$7$F($\"\"\"F*7$F d`mFham-Fi`m6&F[amF*F*F*-%*LINESTYLEG6#\"\"$-F$6%7$7$FhamF(7$FhamFd`mF [bmF]bm-F$6&7#7$FhamFham-%'SYMBOLG6#%'CIRCLEGFh`m-%&STYLEG6#%&POINTG-F $6&Fhbm-F[cm6#%(DIAMONDGFh`mF^cm-F$6&Fhbm-F[cm6#%&CROSSGFh`mF^cm-%+AXE SLABELSG6%Q\"x6\"Q\"yF`dm-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F(Fd`mFidm" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "It is intuitively clear that " }{XPPEDIT 18 0 "g(x)" " 6#-%\"gG6#%\"xG" }{TEXT -1 21 " is discontinuous at " }{XPPEDIT 18 0 " x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 10 ", because " }{XPPEDIT 18 0 "g(x )" "6#-%\"gG6#%\"xG" }{TEXT -1 27 " still \"jumps\" in value as " } {TEXT 306 1 "x" }{TEXT -1 22 " increases through 1. " }}{PARA 0 "" 0 " " {TEXT -1 31 "We can check for continuity at " }{XPPEDIT 18 0 "x = 1; " "6#/%\"xG\"\"\"" }{TEXT -1 68 " with reference to the 3 step analysi s to see whether the condition " }{XPPEDIT 18 0 "Limit(g(x),x = 1) = g (1);" "6#/-%&LimitG6$-%\"gG6#%\"xG/F*\"\"\"-F(6#F," }{TEXT -1 8 " hold s. " }}{PARA 15 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g(1);" "6#-%\"g G6#\"\"\"" }{TEXT -1 12 " exists and " }{XPPEDIT 18 0 "g(1) = 1;" "6#/ -%\"gG6#\"\"\"F'" }{TEXT -1 2 ". " }}{PARA 15 "" 0 "" {XPPEDIT 18 0 "L imit(g(x),x = 1);" "6#-%&LimitG6$-%\"gG6#%\"xG/F)\"\"\"" }{TEXT -1 24 " does not exist because " }{XPPEDIT 18 0 "Limit(g(x),x = 1^`-`) = -in finity;" "6#/-%&LimitG6$-%\"gG6#%\"xG/F*)\"\"\"%\"-G,$%)infinityG!\"\" " }{TEXT -1 8 " while " }{XPPEDIT 18 0 "Limit(g(x),x = 1^`+`) = infin ity;" "6#/-%&LimitG6$-%\"gG6#%\"xG/F*)\"\"\"%\"+G%)infinityG" }{TEXT -1 111 ". \nActually, it is sufficient to note that just one of these \+ two one-sided limits is infinite to conclude that " }{XPPEDIT 18 0 "Li mit(g(x),x = 1)" "6#-%&LimitG6$-%\"gG6#%\"xG/F)\"\"\"" }{TEXT -1 42 " \+ does not exist (as a finite real number)." }}{PARA 15 "" 0 "" {TEXT -1 40 "not applicable because of the 2nd step. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Therefore, " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 22 " is not continuous at " } {XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 3 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "Consider the piecewise fu nction f defined for all real numbers by the piecewise formula: " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)=PIECEWISE([x,x<0 ],[x^2,0<=x and x<1],[2,1<=x])" "6#/-%\"fG6#%\"xG-%*PIECEWISEG6%7$F'2F '\"\"!7$*$F'\"\"#31F-F'2F'\"\"\"7$F01F4F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "The graph of " }{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG- %\"fG6#%\"xG" }{TEXT -1 26 " can be drawn as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 272 "f := x -> piecewise(x<0,x,x<1,x^2,2):\n'f(x)'=f(x);\np1 := plot(f(x),x=-3..3 ,discont=true,thickness=2):\np2 := plot([[1,1]],style=point,color=red, symbol=circle):\np3 := plot([[[1,2]]$2],style=point,color=red,symbol=[ circle,cross]):\nplots[display]([p1,p2,p3],labels=[`x`,`y`]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%*PIECEWISEG6%7$F'2F'\" \"!7$*$)F'\"\"#\"\"\"2F'F27$F1%*otherwiseG" }}{PARA 13 "" 1 "" {GLPLOT2D 315 318 318 {PLOTDATA 2 "6)-%'CURVESG6&7U7$$!\"$\"\"!F(7$$!3 1;TY$Q6G\"H!#'**>F.FN7$$!3`80j^5*H\">F.FQ7$$!3zPdvJ\"3&H=F.FT7$$!3 %oy-Fk(p`9F.F]o7$$!3:>)***HRXL8F.F`o7$$!3wVI7&=/8D\"F.Fco 7$$!3A#G=Wa*el6F.Ffo7$$!3S^j.Zn(o3\"F.Fio7$$!3xr.LhV(>+\"F.F\\p7$$!3Si liRk%y8*!#=F_p7$$!3QtZWdB:q$)FapFcp7$$!3!3!=-<<-TvFapFfp7$$!3'yKt\\j[W o'FapFip7$$!3!=JIi)*ek%eFapF\\q7$$!3E6lW74mN]FapF_q7$$!3D*)oz))ySNTFap Fbq7$$!3pNpn10\\ELFapFeq7$$!3+/eIT&)ziCFapFhq7$$!3?)383Dl,o\"FapF[r7$$ !3)H*=jPMSX#)!#>F^r7$$!38DL+_;-?UF`rFbr7$$!3wsvuj')RY>!#?Fer7$$\"3.yqk gR/8SF`r$\"3d)f3I=_/h\"Fgr7$$\"3K8*owyF2A)F`r$\"3m#=&*e`O!enFgr7$$\"3; )3meyG[k\"Fap$\"3')4P\"[thaq#F`r7$$\"3TI2Cw!yh]#Fap$\"3C)oH)\\&G4G'F`r 7$$\"3!4.jm\")fdL$Fap$\"3pxjWb$HF6\"Fap7$$\"3!yZ!Rlp7%=%Fap$\"3%Gx]i%= p]H/tL)Fap7$$\"2%******R********F.$\"2!******z)*******F.7S7$$\"30+ ++1+++5F.$\"\"#F*7$$\"3b]D?9VfV5F.F]w7$$\"3?!4s')[D:3\"F.F]w7$$\"3<%yg 91$=C6F.F]w7$$\"3;j>L'RBr;\"F.F]w7$$\"3%y3(GV'f)47F.F]w7$$\"3i))[$[h\" [\\7F.F]w7$$\"3Mw%y8(y]!H\"F.F]w7$$\"3a@Xe%GPHL\"F.F]w7$$\"373V7E1Bv8F .F]w7$$\"3;j/TEXt=9F.F]w7$$\"35v@Y&y_qX\"F.F]w7$$\"3!*H%*\\p+>+:F.F]w7 $$\"3w'[p$zW]V:F.F]w7$$\"3QiUCRfC&e\"F.F]w7$$\"3!*zQr$=^Ji\"F.F]w7$$\" 39%*>m&=C#o;F.F]w7$$\"33YuaIpS1 F.F]w7$$\"3+#)p/J;cc>F.F]w7$$\"3)\\HOQ#G,**>F.F]w7$$\"3k5SX#o2J/#F.F]w 7$$\"3gAb]'Q#\\\"3#F.F]w7$$\"3v'[k%=*[H7#F.F]w7$$\"3MnE]svxl@F.F]w7$$ \"33pp([0xw?#F.F]w7$$\"33\\`]ep@[AF.F]w7$$\"3I6.i4'HKH#F.F]w7$$\"3&*Rc myanLBF.F]w7$$\"3!*>%po2goP#F.F]w7$$\"3Ve`LT<*fT#F.F]w7$$\"3;\"oBm)Hxe CF.F]w7$$\"3C'e>W!o-*\\#F.F]w7$$\"3J*oEEk.6a#F.F]w7$$\"3GU*>HWTAe#F.F] w7$$\"3>tSP2*3`i#F.F]w7$$\"3apHL%*zymEF.F]w7$$\"3L9dq^j?4FF.F]w7$$\"3? YGmjMF^FF.F]w7$$\"3g8-jq(G**y#F.F]w7$$\"3SqRm9@BMGF.F]w7$$\"3)4w6PbdQ( GF.F]w7$$\"3]!o,l`1h\"HF.F]w7$$\"3wn.)Q?Wl&HF.F]w7$$\"\"$F*F]w-%'COLOU RG6&%$RGBG$\"*++++\"!\")$F*F*Ff`l-%*THICKNESSG6#F^w-%'POINTSG6$7$$\"\" \"F*F]wF_`l-F$6&7#7$F^alF^alF_`l-%&STYLEG6#%&POINTG-%'SYMBOLG6#%'CIRCL EG-F$6&7#F]alFhalF_`lFdal-F$6&F^bl-Fial6#%&CROSSGF_`lFdal-%+AXESLABELS G6%Q\"x6\"Q!Fhbl-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F(F]`lF]cl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curv e 3" "Curve 4" "Curve 5" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 29 "It is intuitively clear that " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 21 " is discontinuous at " } {XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 10 ", because " } {XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 46 " \"jumps\" from \+ values near 1 to the value 2 as " }{TEXT 307 1 "x" }{TEXT -1 22 " incr eases through 1. " }}{PARA 0 "" 0 "" {TEXT -1 36 "There is a \"break\" in the graph of " }{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG-%\"fG6#%\"xG" } {TEXT -1 7 " where " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 11 ". Indeed " }{XPPEDIT 18 0 "f(x)->1" "6#f*6#-%\"fG6#%\"xG7\"6$%)o peratorG%&arrowG6\"\"\"\"F-F-F-" }{TEXT -1 5 ", as " }{XPPEDIT 18 0 "x ->1" "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\"\"\"\"F*F*F*" }{XPPEDIT 18 0 "``^`-`" "6#)%!G%\"-G" }{TEXT -1 8 ", while " }{XPPEDIT 18 0 "f(x )->2" "6#f*6#-%\"fG6#%\"xG7\"6$%)operatorG%&arrowG6\"\"\"#F-F-F-" } {TEXT -1 4 " as " }{XPPEDIT 18 0 "x->1" "6#f*6#%\"xG7\"6$%)operatorG%& arrowG6\"\"\"\"F*F*F*" }{XPPEDIT 18 0 "``^`+` " "6#)%!G%\"+G" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 80 "The function appears to be co ntinuous at all other real number values including " }{XPPEDIT 18 0 "x =0" "6#/%\"xG\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 31 "We can check for continuity at " } {XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 68 " with reference \+ to the 3 step analysis to see whether the condition " }{XPPEDIT 18 0 " Limit(f(x),x = 1) = f(1);" "6#/-%&LimitG6$-%\"fG6#%\"xG/F*\"\"\"-F(6#F ," }{TEXT -1 8 " holds. " }}{PARA 15 "" 0 "" {TEXT -1 7 " Since " } {XPPEDIT 18 0 "f(x) = 2;" "6#/-%\"fG6#%\"xG\"\"#" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "2 <= x;" "6#1\"\"#%\"xG" }{TEXT -1 18 ", it follows th at " }{XPPEDIT 18 0 "f(1) = 2;" "6#/-%\"fG6#\"\"\"\"\"#" }{TEXT -1 2 " . " }}{PARA 15 "" 0 "" {XPPEDIT 18 0 "Limit(f(x),x = 1);" "6#-%&LimitG 6$-%\"fG6#%\"xG/F)\"\"\"" }{TEXT -1 25 " does not exist because " } {XPPEDIT 18 0 "Limit(f(x),x = 1^`-`) = Limit(x^2,x = 1^`-`);" "6#/-%&L imitG6$-%\"fG6#%\"xG/F*)\"\"\"%\"-G-F%6$*$F*\"\"#/F*)F-F." }{XPPEDIT 18 0 "`` = 1;" "6#/%!G\"\"\"" }{TEXT -1 9 ", while " }{XPPEDIT 18 0 " Limit(f(x),x = 1^`+`) = Limit(2,x = 1^`+`);" "6#/-%&LimitG6$-%\"fG6#% \"xG/F*)\"\"\"%\"+G-F%6$\"\"#/F*)F-F." }{XPPEDIT 18 0 "`` = 2;" "6#/%! G\"\"#" }{TEXT -1 2 ". " }}{PARA 15 "" 0 "" {TEXT -1 40 "not applicabl e because of the 2nd step. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Therefore, f is not continuous at " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "We can check for continui ty at " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 68 " with refer ence to the 3 step analysis to see whether the condition " }{XPPEDIT 18 0 "Limit(f(x),x = 0) = f(0);" "6#/-%&LimitG6$-%\"fG6#%\"xG/F*\"\"!- F(6#F," }{TEXT -1 8 " holds. " }}{PARA 15 "" 0 "" {TEXT -1 7 " Since \+ " }{XPPEDIT 18 0 "f(x)=x^2" "6#/-%\"fG6#%\"xG*$F'\"\"#" }{TEXT -1 5 " \+ for " }{XPPEDIT 18 0 "0<=x" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "``<1" "6#2 %!G\"\"\"" }{TEXT -1 18 ", it follows that " }{XPPEDIT 18 0 "f(0)=0^2 " "6#/-%\"fG6#\"\"!*$F'\"\"#" }{XPPEDIT 18 0 "``=0" "6#/%!G\"\"!" } {TEXT -1 3 ". " }}{PARA 15 "" 0 "" {XPPEDIT 18 0 "Limit(f(x),x = 0) = 0;" "6#/-%&LimitG6$-%\"fG6#%\"xG/F*\"\"!F," }{TEXT -1 10 " because \+ " }{XPPEDIT 18 0 "Limit(f(x),x = 0^`-`) = Limit(x,x = 0^`-`);" "6#/-%& LimitG6$-%\"fG6#%\"xG/F*)\"\"!%\"-G-F%6$F*/F*)F-F." }{XPPEDIT 18 0 " ` `=0" "6#/%!G\"\"!" }{TEXT -1 8 " while " }{XPPEDIT 18 0 "Limit(f(x),x = 0^`+`) = Limit(x^2,x = 0^`+`);" "6#/-%&LimitG6$-%\"fG6#%\"xG/F*)\" \"!%\"+G-F%6$*$F*\"\"#/F*)F-F." }{XPPEDIT 18 0 "``=0" "6#/%!G\"\"!" } {TEXT -1 7 " also. " }}{PARA 15 "" 0 "" {XPPEDIT 18 0 "Limit(f(x),x = \+ 0)=0" "6#/-%&LimitG6$-%\"fG6#%\"xG/F*\"\"!F," }{XPPEDIT 18 0 "``=f(0) " "6#/%!G-%\"fG6#\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Therefore, f " }{TEXT 262 2 "is" } {TEXT -1 15 " continuous at " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Examp le 4 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 " Consider the function f defined for all real numbers by the piecewise \+ formula: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = P IECEWISE([(x^3+1)/(x+1), x <> -1], [2, x = -1])" "6#/-%\"fG6#%\"xG-%*P IECEWISEG6$7$*&,&*$F'\"\"$\"\"\"F0F0F0,&F'F0F0F0!\"\"0F',$F0F27$\"\"#/ F',$F0F2" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 14 "The graph of " }{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 26 " can be drawn as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 303 "f := x -> piecewise(x<>-1,(x^3+1)/ (x+1),x=-1,2):\n'f(x)'=f(x);\np1 := plot(f(x),x=-2.2..3.2,0..8,thickne ss=1):\np2 := plot([[-1,3]],style=point,color=red,symbol=circle,symbol size=10):\np3 := plot([[[-1,2]]$3],style=point,color=red,symbol=[circl e,diamond,cross]):\nplots[display]([p1,p2,p3],labels=[`x`,`y`]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$*&,&*$) F'\"\"$\"\"\"F1F1F1F1,&F1F1F'F1!\"\"0F'F37$\"\"#/F'F3" }}{PARA 13 "" 1 "" {GLPLOT2D 404 412 412 {PLOTDATA 2 "6)-%'CURVESG6%7S7$$!3;+++++++A !#<$\"3#4++++++/)F*7$$!3+++]n`H#3#F*$\"3kf>,l$\\#=uF*7$$!3C+]7'>\"))z> F*$\"3!>O%*ou5)**oF*7$$!35++D\\dqk=F*$\"3[/g`!GL=M'F*7$$!3+++vWow[WF*7$$!3<+]7X$p5I\"F*$\"3c (**[fy]Q*RF*7$$!3A+]iAt(o=\"F*$\"3/*['e,^b&f$F*7$$!39++]\"z;%p5F*$\"3? b@Wl!pI@$F*7$$!3I-+]<\\df'*!#=$\"3?]zhn8.**GF*7$$!39+++X#o[\\)F\\o$\"3 ]$4*\\*o96d#F*7$$!3\">++]'F \\o$\"3#o.C?hKS+#F*7$$!3)3+]7:=\\<&F\\o$\"3Knj%Qf*G&y\"F*7$$!3k+++&4Zz &RF\\o$\"3a*=o:;[Cb\"F*7$$!35,++!)H,FHF\\o$\"3)[)3&yMv$y8F*7$$!3Y***\\ 7_\"zF$\"3m>Z!y)p1r5F*7$$\"3A2+ ](y>P)\\Feq$\"3!yJqT[lk_*F\\o7$$\"39***\\i`$R2;F\\o$\"3eGyyhy(4l)F\\o7 $$\"3a-+](=TXw#F\\o$\"3i*QQ-hF(**zF\\o7$$\"3H,+v`R;FQF\\o$\"3?L$Q\"RWb PwF\\o7$$\"3i)***\\ihMt\\F\\o$\"3TZ`qU52+vF\\o7$$\"3e***\\([t!R;'F\\o$ \"3\\DukJ!oaj(F\\o7$$\"3A***\\7O%H+sF\\o$\"3#Q&[hF&HT)zF\\o7$$\"3:,+]F 2i>$)F\\o$\"3ujZWx\"))>g)F\\o7$$\"3m(******Q%*fZ*F\\o$\"3\"F*7$$\"3q***\\-%*>F*$\"3SBXon?Y%)GF*7$$\"3K*** *\\#*=0s?F*$\"3BHGr_rM@KF*7$$\"35+](o/M$)=#F*$\"3F:t$Q&=Z+OF*7$$\"3V++ +#eF.I#F*$\"3Go*4DSz6*RF*7$$\"3A++DZr&[T#F*$\"3;4WP%*yn;WF*7$$\"3\\+]( )\\$Q%GDF*$\"39:OCR@ck[F*7$$\"3!*******yw!Gj#F*$\"3-n8g&fo))H&F*7$$\"3 !)***\\#3nU_FF*$\"3W69ooF*7$$\"3?+]7]$pE3$F*$\"3'34v?(4=?uF*7$$\"3;++ +++++KF*F+-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!F_[lF^[l-%*THICKNESSG6#\" \"\"-F$6&7#7$$F][lF_[l$\"\"$F_[l-Fhz6&Fjz$\"*++++\"!\")F^[lF^[l-%'SYMB OLG6$%'CIRCLEGF\\[l-%&STYLEG6#%&POINTG-F$6&7#7$Fh[l$\"\"#F_[l-Fa\\l6#F c\\lF[\\lFd\\l-F$6&Fj\\l-Fa\\l6#%(DIAMONDGF[\\lFd\\l-F$6&Fj\\l-Fa\\l6# %&CROSSGF[\\lFd\\l-%+AXESLABELSG6%%\"xG%\"yG-%%FONTG6#%(DEFAULTG-%%VIE WG6$;$!#AF][l$\"#KF][l;F^[l$\"\")F_[l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "It is intuitively clear that " }{XPPEDIT 18 0 "f(x); " "6#-%\"fG6#%\"xG" }{TEXT -1 21 " is discontinuous at " }{XPPEDIT 18 0 "x = -1;" "6#/%\"xG,$\"\"\"!\"\"" }{TEXT -1 14 ", because, as " } {TEXT 309 1 "x" }{TEXT -1 19 " increases through " }{XPPEDIT 18 0 "-1 " "6#,$\"\"\"!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6 #%\"xG" }{TEXT -1 46 " \"jumps\" from values near 3 to the value 2 at \+ " }{XPPEDIT 18 0 "x = -1;" "6#/%\"xG,$\"\"\"!\"\"" }{TEXT -1 42 ", and then \"jumps\" back to values near 3. " }}{PARA 0 "" 0 "" {TEXT -1 36 "There is a \"break\" in the graph of " }{XPPEDIT 18 0 "y=f(x)" "6 #/%\"yG-%\"fG6#%\"xG" }{TEXT -1 7 " where " }{XPPEDIT 18 0 "x = -1;" " 6#/%\"xG,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 68 "The function appears to be continuous at all non-zero real numbers. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "We ca n check for continuity at " }{XPPEDIT 18 0 "x = -1;" "6#/%\"xG,$\"\"\" !\"\"" }{TEXT -1 68 " with reference to the 3 step analysis to see whe ther the condition " }{XPPEDIT 18 0 "Limit(f(x),x = -1) = f(-1);" "6#/ -%&LimitG6$-%\"fG6#%\"xG/F*,$\"\"\"!\"\"-F(6#,$F-F." }{TEXT -1 8 " hol ds. " }}{PARA 15 "" 0 "" {TEXT -1 50 " The piecewise formula for f giv es the value 2 at " }{XPPEDIT 18 0 "x = -1;" "6#/%\"xG,$\"\"\"!\"\"" } {TEXT -1 11 ", that is, " }{XPPEDIT 18 0 "f(-1) = 2;" "6#/-%\"fG6#,$\" \"\"!\"\"\"\"#" }{TEXT -1 2 ". " }}{PARA 258 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Limit(f(x),x=-1)=Limit((x^3+1)/(x+1),x=-1)" "6#/-%&Limi tG6$-%\"fG6#%\"xG/F*,$\"\"\"!\"\"-F%6$*&,&*$F*\"\"$F-F-F-F-,&F*F-F-F-F ./F*,$F-F." }{TEXT -1 3 " \n " }{XPPEDIT 18 0 "``=Limit((x+1)*(x^2-x+1 )/(x+1),x=-1)" "6#/%!G-%&LimitG6$*(,&%\"xG\"\"\"F+F+F+,(*$F*\"\"#F+F*! \"\"F+F+F+,&F*F+F+F+F//F*,$F+F/" }{TEXT -1 3 " \n " }{XPPEDIT 18 0 "`` =Limit(x^2-x+1,x=-1)" "6#/%!G-%&LimitG6$,(*$%\"xG\"\"#\"\"\"F*!\"\"F,F ,/F*,$F,F-" }{TEXT -1 3 " \n " }{XPPEDIT 18 0 "``=3" "6#/%!G\"\"$" } {TEXT -1 2 ". " }}{PARA 15 "" 0 "" {XPPEDIT 18 0 "Limit(f(x),x = -1) < > f(-1);" "6#0-%&LimitG6$-%\"fG6#%\"xG/F*,$\"\"\"!\"\"-F(6#,$F-F." } {TEXT -1 10 " because " }{XPPEDIT 18 0 "Limit(f(x),x = -1) = 3;" "6#/ -%&LimitG6$-%\"fG6#%\"xG/F*,$\"\"\"!\"\"\"\"$" }{TEXT -1 8 " while " }{XPPEDIT 18 0 "f(-1) = 2;" "6#/-%\"fG6#,$\"\"\"!\"\"\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 " Therefore, f is not continuous at " }{XPPEDIT 18 0 "x = -1;" "6#/%\"x G,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 261 4 "Note" }{TEXT -1 63 ": The function f can \+ also be defined by the (simpler) formula: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = PIECEWISE([x^2-x+1, x <> -1],[2, x = - 1]);" "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$,(*$F'\"\"#\"\"\"F'!\"\"F/F/0F ',$F/F07$F./F',$F/F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 10 "Example 5 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 79 "Consider the function f defined for all r eal numbers by the piecewise formula: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = PIECEWISE([2-sqrt(-x), x < 0],[1, x = 0], [2-sqrt(x), 0 < x]);" "6#/-%\"fG6#%\"xG-%*PIECEWISEG6%7$,&\"\"#\"\"\"- %%sqrtG6#,$F'!\"\"F32F'\"\"!7$F./F'F57$,&F-F.-F06#F'F32F5F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 14 "The graph of " }{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 16 " is as follows. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 300 "f := x -> piecewise(x<0,2-sqrt(-x),x=0,1,x>0,2-sqrt(x)):\n'f(x)'= f(x);\np1 := plot(f(x),x=-6..6,thickness=2):\np2 := plot([[0,2]],style =point,color=red,symbol=circle,symbolsize=10):\np3 := plot([[[0,1]]$3] ,style=point,color=red,symbol=[circle,diamond,cross]):\nplots[display] ([p1,p2,p3],labels=[`x`,`y`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% \"fG6#%\"xG-%*PIECEWISEG6%7$,&\"\"#\"\"\"*$,$F'!\"\"#F.F-F12F'\"\"!7$F ./F'F47$,&F-F.*$F'F2F12F4F'" }}{PARA 13 "" 1 "" {GLPLOT2D 625 178 178 {PLOTDATA 2 "6)-%'CURVESG6%7en7$$!\"'\"\"!$!3!)y<$yU(*[\\%!#=7$$!3z*** ***\\TVQd!#<$!3-PYr7!H]&RF-7$$!3l****\\-r%3^&F1$!3[@YD3N>vMF-7$$!3A+++ l;!\\D&F1$!3yR(R#*GsN#HF-7$$!3o*****\\lfs*\\F1$!3k`$zuL^XN#F-7$$!3%)** **\\s@%3u%F1$!3g?tP%3vMx\"F-7$$!3J++]U.6.XF1$!3q'p8oJL0A\"F-7$$!3')*** *\\-G&pD%F1$!3\")HOW*fUQK'!#>7$$!3(*****\\AjP-SF1$!3=_'=D=C*Rf!#@7$$!3 3++]sih[PF1$\"35pW[wOc'Q'FR7$$!3%)******pGf([$F1$\"3Oz>ua>!\\K\"F-7$$! 3)******\\J$odKF1$\"3'o3i1_Y4&>F-7$$!3y******4'f))*HF1$\"3KDsefUy#o#F- 7$$!33+++]J(*QFF1$\"3w>qEZm:]MF-7$$!3#)*******QC&)[#F1$\"3KWn1/D%\\A%F -7$$!3/++]AH4hAF1$\"33@P#[0pI'\\F-7$$!3%*******4\\l!*>F1$\"3om`LWH%4*e F-7$$!3'*******R%e:w\"F1$\"3B4!)[Y&Gws'F-7$$!33++]#yk]\\\"F1$\"3GS%*)o #ersxF-7$$!3M+++SF)***F-7$$!3(p****\\ZD\"RvF-$\"3>3_BC'=<8\"F17$$!3I'*****\\ion\\F-$ \"3IZXp\\==&H\"F17$$!3A'***\\(G%*py$F-$\"31&4\\kS9YQ\"F17$$!3:'****\\K -jg#F-$\"3O?m()G/[*[\"F17$$!3Q(**\\7VM&p>F-$\"3'32$oHc?c:F17$$!3j)*** \\PlwK8F-$\"3q@cH[#H\\j\"F17$$!3o)***\\Pk)*fpFR$\"3#[Gxy8#=OF17$$\"3I*)**\\78#>-'FR$\"3]R+xmRgaF-$\"3m^!eAYZ7c\"F17$$ \"3/*****\\xgke#F-$\"3uy*oam%**F-$\"3Fdytc3n-5F17$$\"3k*****\\JigC\"F1 $\"35>VZlQGP))F-7$$\"3%*****\\PIFU,UF-7$$\" 3q+++5zj_FF1$\"3718vcV#*3MF-7$$\"3=****\\<3;%*HF1$\"3k`)zCScjp#F-7$$\" 3;++]Z=iYKF1$\"3w`#3zO:;)>F-7$$\"3[******\\'[M\\$F1$\"3&3S\"o'[I#48F-7 $$\"3W****\\PM&=v$F1$\"3(4A[30$)HI'FR7$$\"3v+++gzs+SF1$!3fbTj1s\")>=FX 7$$\"35+++0\"Q_D%F1$!3ad37n[G#G'FR7$$\"3q++]x2k2XF1$!3y2_b#G07B\"F-7$$ \"3d+++?EdRZF1$!3Uz\\FC&f0x\"F-7$$\"3M+++&o#R0]F1$!3a84jN\\tsBF-7$$\"3 ++++?`9V_F1$!3-1](4X:z*GF-7$$\"3G++]<#Rm\\&F1$!33Z#R=\\7\\W$F-7$$\"3F+ +]A_ERdF1$!3a$zH8gjn&RF-7$$\"\"'F*F+-%'COLOURG6&%$RGBG$\"#5!\"\"$F*F*F i]l-%*THICKNESSG6#\"\"#-F$6&7#7$Fi]l$F]^lF*-Fc]l6&Fe]l$\"*++++\"!\")Fi ]lFi]l-%&STYLEG6#%&POINTG-%'SYMBOLG6$%'CIRCLEGFg]l-F$6&7#7$Fi]l$\"\"\" F*-F]_l6#F__lFc^lFh^l-F$6&Fb_l-F]_l6#%(DIAMONDGFc^lFh^l-F$6&Fb_l-F]_l6 #%&CROSSGFc^lFh^l-%+AXESLABELSG6%%\"xG%\"yG-%%FONTG6#%(DEFAULTG-%%VIEW G6$;F(F`]lFj`l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "It is int uitively clear that " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 21 " is discontinuous at " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 14 ", because, as " }{TEXT 308 1 "x" }{TEXT -1 22 " increase s through 0, " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 46 " \"jumps\" from values near 2 to the value 1 at " }{XPPEDIT 18 0 "x=0 " "6#/%\"xG\"\"!" }{TEXT -1 42 ", and then \"jumps\" back to values ne ar 2. " }}{PARA 0 "" 0 "" {TEXT -1 36 "There is a \"break\" in the gra ph of " }{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 7 " where " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 68 "The function appears to be continuous at \+ all non-zero real numbers. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "We can check for continuity at " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 68 " with reference to the 3 step analysis to see whether the condition " }{XPPEDIT 18 0 "Limit(f(x),x \+ = 0) = f(0);" "6#/-%&LimitG6$-%\"fG6#%\"xG/F*\"\"!-F(6#F," }{TEXT -1 8 " holds. " }}{PARA 15 "" 0 "" {TEXT -1 50 " The piecewise formula fo r f gives the value 1 at " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" } {TEXT -1 11 ", that is, " }{XPPEDIT 18 0 "f(0) = 1;" "6#/-%\"fG6#\"\"! \"\"\"" }{TEXT -1 2 ". " }}{PARA 15 "" 0 "" {XPPEDIT 18 0 "Limit(f(x), x = 0) = 2;" "6#/-%&LimitG6$-%\"fG6#%\"xG/F*\"\"!\"\"#" }{TEXT -1 11 " because " }{XPPEDIT 18 0 "Limit(f(x),x = 0^`-`) = Limit(2-sqrt(-x), x = 0^`-`);" "6#/-%&LimitG6$-%\"fG6#%\"xG/F*)\"\"!%\"-G-F%6$,&\"\"#\" \"\"-%%sqrtG6#,$F*!\"\"F8/F*)F-F." }{XPPEDIT 18 0 "`` = 2;" "6#/%!G\" \"#" }{TEXT -1 9 ", while " }{XPPEDIT 18 0 "Limit(f(x),x = 0^`+`) = L imit(2-sqrt(x),x = 0^`+`);" "6#/-%&LimitG6$-%\"fG6#%\"xG/F*)\"\"!%\"+G -F%6$,&\"\"#\"\"\"-%%sqrtG6#F*!\"\"/F*)F-F." }{XPPEDIT 18 0 "`` = 2;" "6#/%!G\"\"#" }{TEXT -1 2 ". " }}{PARA 15 "" 0 "" {XPPEDIT 18 0 "Limit (f(x),x = 0) <> f(0)" "6#0-%&LimitG6$-%\"fG6#%\"xG/F*\"\"!-F(6#F," } {TEXT -1 10 " because " }{XPPEDIT 18 0 "Limit(f(x),x = 0) = 2" "6#/-% &LimitG6$-%\"fG6#%\"xG/F*\"\"!\"\"#" }{TEXT -1 8 " while " }{XPPEDIT 18 0 "f(0)=1" "6#/-%\"fG6#\"\"!\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Therefore, f is not continuous at " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 4 "Not e" }{TEXT -1 53 ": The function f can also be defined by the formula: \+ " }}{PARA 260 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = PIECEWISE( [2-sqrt(abs(x)), x <> 0],[1, x = 0]);" "6#/-%\"fG6#%\"xG-%*PIECEWISEG6 $7$,&\"\"#\"\"\"-%%sqrtG6#-%$absG6#F'!\"\"0F'\"\"!7$F./F'F7" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 " " {TEXT -1 17 "Code for picture " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 29 "Graph of continuous \+ function " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 458 "f := x ->2+(x+2)^3/3: \np1 := plot(f(x),x=0.5..1.5,t hickness=2,color=red):\np2 := plot([[0,0],[2,0]],color=black):\np3 := \+ plot([[1,0],[1,f(1)]],color=black,linestyle=2):\np4 := plot([[[1,f(1)] ]$3],style=point,symbol=[circle,diamond,cross],color=red):\nt1 := plot s[textplot]([[1,-.6,`x = a`],[0.85,12,`( a,f(a) )`]],color=COLOR(RGB,. 01,0,0)):\nt2 := plots[textplot]([[1.45,13,`y = f(x)`]],color=red):\np lots[display]([p||(1..4),t1,t2],view=[0..2,-.6..17],axes=none);" }} {PARA 13 "" 1 "" {GLPLOT2D 348 263 263 {PLOTDATA 2 "6.-%'CURVESG6%7S7$ $\"3++++++++]!#=$\"3/LLLLLL3s!#<7$$\"35mmmT:(z@&F*$\"3-,yv)ycdM(F-7$$ \"3jLLe9ui2aF*$\"31wWJZqFnuF-7$$\"3Anm;z_\"4i&F*$\"3&\\pHCdBhg(F-7$$\" 3$pmmT&phNeF*$\"3))y=Z8[C[xF-7$$\"35LLe*=)H\\gF*$\"3C`0'3Zb?*yF-7$$\"3 ;nm\"z/3uC'F*$\"3uK\\OM5^F!)F-7$$\"37++DJ$RDX'F*$\"3uDzrM(Q*p\")F-7$$ \"3'fm;zR'okmF*$\"3cp'zxcz&>$)F-7$$\"3I++D1J:woF*$\"3aU'*p3\"H6Z)F-7$$ \"3WLLL3En$4(F*$\"30ou]=X_H')F-7$$\"3qmm;/RE&G(F*$\"3e?NLiA;r()F-7$$\" 3\")*****\\K]4](F*$\"3nORyu.,L*)F-7$$\"3$******\\PAvr(F*$\"3!=$=2$e(4) 4*F-7$$\"3`+++v'Hi#zF*$\"3_\"G;Ck\\'f#*F-7$$\"3jmm\"z*ev:\")F*$\"3r/AI e;Y3%*F-7$$\"3kKLL347T$)F*$\"3Q<&zS`V!)e*F-7$$\"3,LLLLY.K&)F*$\"38`&)3 *QCCu*F-7$$\"3?***\\7o7Tv)F*$\"3K`RtKQiC**F-7$$\"3IKLL$Q*o]*)F*$\"35q` E\"pE)35!#;7$$\"3A++D\"=lj;*F*$\"3nx!yiRQq-\"Fgq7$$\"3]***\\PaR#\\ECjW5Fgq7$$\"3!HLLe9Ege*F*$\"3#=5k86aK1\"Fgq7$$\"3GLLeR\"3Gy* F*$\"3)*40'z!Rf!3\"Fgq7$$\"3cmm;/T1&***F*$\"3c%>gC%eb*4\"Fgq7$$\"3em;z RQb@5F-$\"3^x&*)*f\"Q&>6Fgq7$$\"3%)**\\(=>Y2/\"F-$\"3(fG)H.>\"Fgq7$$\"3#***\\7y%3T7\"F- $\"3r30pDAQ;7Fgq7$$\"3#****\\P![hY6F-$\"3%HC([Zr]Q7Fgq7$$\"3ELLLQx$o; \"F-$\"3K3VX5#f'e7Fgq7$$\"3')****\\P+V)=\"F-$\"33f-$*4=Y!G\"Fgq7$$\"3i m;zpe*z?\"F-$\"3RsBB8\\Z+8Fgq7$$\"3)*****\\#\\'QH7F-$\"3:LP*>iNEK\"Fgq 7$$\"37L$e9S8&\\7F-$\"3(Hvs0%pvV8Fgq7$$\"3%***\\i?=bq7F-$\"3=q^&eD;hO \"Fgq7$$\"3GLL$3s?6H\"F-$\"3]#zz$fkD)Q\"Fgq7$$\"3&***\\7`Wl78F-$\"3m[7 h6Ot69Fgq7$$\"3emmm'*RRL8F-$\"3#*e\\;1`jM9Fgq7$$\"3_mmTvJga8F-$\"3y*G> >=_$e9Fgq7$$\"3KL$e9tOcP\"F-$\"3/epEE2<#[\"Fgq7$$\"3'******\\Qk\\R\"F- $\"3je#f'z2K/:Fgq7$$\"3@LL3dg6<9F-$\"3%)fZmo#>+`\"Fgq7$$\"3_mmmw(GpV\" F-$\"3'*)G2wM)G`:Fgq7$$\"3-+]7oK0e9F-$\"3+CY\"[H&Ry:Fgq7$$\"3-+](=5s#y 9F-$\"3B\\E3&)[r-;Fgq7$$\"3++++++++:F-$\"3Vmmmmm;H;Fgq-%'COLOURG6&%$RG BG$\"*++++\"!\")$\"\"!Fa[lF`[l-%*THICKNESSG6#\"\"#-F$6$7$7$F`[lF`[l7$$ Fe[lFa[lF`[l-Fjz6&F\\[lFa[lFa[lFa[l-F$6%7$7$$\"\"\"Fa[lF`[l7$Fb\\l$\"# 6Fa[lF\\\\l-%*LINESTYLEGFd[l-F$6&7#Fd\\l-%'SYMBOLG6#%'CIRCLEGFiz-%&STY LEG6#%&POINTG-F$6&F[]l-F]]l6#%(DIAMONDGFizF`]l-F$6&F[]l-F]]l6#%&CROSSG FizF`]l-%%TEXTG6%7$Fb\\l$!\"'!\"\"Q&x~=~a6\"-%&COLORG6&F\\[l$Fc\\l!\"# F`[lF`[l-F_^l6%7$$\"#&)F[_l$\"#7Fa[lQ+(~a,f(a)~)Ff^lFg^l-F_^l6%7$$\"$X \"F[_l$\"#8Fa[lQ)y~=~f(x)Ff^lFiz-%+AXESLABELSG6%Q\"xFf^lQ!Ff^l-%%FONTG 6#%(DEFAULTG-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$;F`[lF[\\l;Fb^l$\"# " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }}{MARK "10" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }