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"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 "Norm al" -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 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 31 "More on continuity of functions" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 18 "Version: 22.3.2007" }}{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 of a function " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "The limit of a funct ion " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 4 " as " } {TEXT 263 1 "x" }{TEXT -1 21 " approaches a number " }{TEXT 288 1 "c" }{TEXT -1 52 " can often by found simply by calculating the value " } {XPPEDIT 18 0 "f(c);" "6#-%\"fG6#%\"cG" }{TEXT -1 17 " of the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 4 " at " } {XPPEDIT 18 0 "x = c;" "6#/%\"xG%\"cG" }{TEXT -1 10 ", so that " }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Limit(f(x),x = c) = f(c);" "6#/-%&LimitG6$-%\"fG6#%\"xG /F*%\"cG-F(6#F," }{TEXT -1 13 " ------- (i)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "When this happens we say that t he function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 4 " is " }{TEXT 261 10 "continuous" }{TEXT -1 4 " at " }{XPPEDIT 18 0 "x = c ;" "6#/%\"xG%\"cG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 43 "If \+ a function is not continuous at a point " }{XPPEDIT 18 0 "x=c" "6#/%\" xG%\"cG" }{TEXT -1 53 ", then it is said to be discontinuous at that p oint. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Roughly, a function f is continuous at " }{XPPEDIT 18 0 "x=c" "6#/%\" xG%\"cG" }{TEXT -1 29 " provided that the values of " }{XPPEDIT 18 0 " f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 21 " change gradually as " }{TEXT 265 1 "x" }{TEXT -1 29 " increases through the value " }{XPPEDIT 18 0 "x=c" "6#/%\"xG%\"cG" }{TEXT -1 51 " with no sudden jump in value. Con tinuity of f at " }{XPPEDIT 18 0 "x=c" "6#/%\"xG%\"cG" }{TEXT -1 29 " corresponds to the graph of " }{XPPEDIT 18 0 "y = f(x);" "6#/%\"yG-% \"fG6#%\"xG" }{TEXT -1 24 " going through the point" }{XPPEDIT 18 0 "` `(c,f(c))" "6#-%!G6$%\"cG-%\"fG6#F&" }{TEXT -1 16 " with no break. 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c\\mF`\\nQ\"bFgjmFhjm-F_jm6%7$$Ff^mF[\\m$\"#bF[\\mQ)y~=~f(x)FgjmFhz-%* AXESTICKSG6$F`[lF`[l-%+AXESLABELSG6%FfjmQ!Fgjm-%%FONTG6#%(DEFAULTG-%%V IEWG6$;$F_[nF[\\mFbjm;Fa^nF`[n" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Cur ve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 1 8" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 24 " f is not continuous at " }{XPPEDIT 18 0 "x=c" "6#/%\"xG%\"cG" } {TEXT -1 9 " because " }{XPPEDIT 18 0 "Limit(f(x),x=c)" "6#-%&LimitG6$ -%\"fG6#%\"xG/F)%\"cG" }{TEXT -1 16 " does not exist." }}{PARA 0 "" 0 "" {TEXT -1 42 "The picture suggests that the right limit " }{XPPEDIT 18 0 "Limit(f(x),x=c^`+`)" "6#-%&LimitG6$-%\"fG6#%\"xG/F))%\"cG%\"+G" }{TEXT -1 26 " exists and has the value " }{XPPEDIT 18 0 "R = f(c);" " 6#/%\"RG-%\"fG6#%\"cG" }{TEXT -1 68 " and that this value is different from the value of the left limit " }{XPPEDIT 18 0 "Limit(f(x),x = c^ `-`) = L;" "6#/-%&LimitG6$-%\"fG6#%\"xG/F*)%\"cG%\"-G%\"LG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 22 "This discontinuity is " }{TEXT 261 12 "nonremovable" }{TEXT -1 44 " because we cannot modify the defi nition of " }{XPPEDIT 18 0 "f(c)" "6#-%\"fG6#%\"cG" }{TEXT -1 50 " so \+ as to obtain a function that is continuous at " }{XPPEDIT 18 0 "x=c" " 6#/%\"xG%\"cG" }{TEXT -1 2 ". 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zFg^m-%%TEXTG6%7$$\"#AF[\\m$!\"#F[\\mQ\"x6\"-Fg[m6&F[[l$F\\]mFejmF_[lF _[l-F_jm6%7$$!\"&Fejm$FehmF`[lQ\"yFgjmFhjm-F_jm6%7$$F[\\mF[\\m$\"$9'Fe jmQ\"LFgjmFhjm-F_jm6%7$Fe[nF]^mQ%f(c)FgjmFhjm-F_jm6%7$Fi[m$!\"$F[\\mQ \"aFgjmFhjm-F_jm6%7$F[]mF`\\nQ\"cFgjmFhjm-F_jm6%7$Fc\\mF`\\nQ\"bFgjmFh jm-F_jm6%7$$Ff^mF[\\m$\"#lF[\\mQ)y~=~f(x)FgjmFhz-%*AXESTICKSG6$F`[lF`[ l-%+AXESLABELSG6%FfjmQ!Fgjm-%%FONTG6#%(DEFAULTG-%%VIEWG6$;$F_[nF[\\mFb jm;Fa^nF`[n" 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" "Curve 7" " Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve \+ 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" }}{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 24 " f is not continuous at " }{XPPEDIT 18 0 "x=c" "6#/%\"xG% \"cG" }{TEXT -1 19 " because, although " }{XPPEDIT 18 0 "f(c)" "6#-%\" fG6#%\"cG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Limit(f(x),x=c)" "6#-%& LimitG6$-%\"fG6#%\"xG/F)%\"cG" }{TEXT -1 49 " both exist, their values are different, that is " }{XPPEDIT 18 0 "Limit(f(x),x = c)<>f(c)" "6# 0-%&LimitG6$-%\"fG6#%\"xG/F*%\"cG-F(6#F," }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 22 "This discontinuity is " }{TEXT 261 9 "removable" } {TEXT -1 20 " because redefining " }{XPPEDIT 18 0 "f(c)" "6#-%\"fG6#% \"cG" }{TEXT -1 17 " to be the value " }{TEXT 290 1 "L" }{TEXT -1 4 " \+ of " }{XPPEDIT 18 0 "Limit(f(x),x = c)" "6#-%&LimitG6$-%\"fG6#%\"xG/F) %\"cG" }{TEXT -1 40 " gives a function that is continuous at " } {XPPEDIT 18 0 "x=c" "6#/%\"xG%\"cG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT 261 4 "Note" }{TEXT -1 50 ": In order for a discontinuity of a function f at " }{XPPEDIT 18 0 "x=c" "6#/%\"xG%\"cG" }{TEXT -1 39 " t o be removable it is sufficient that " }{XPPEDIT 18 0 "Limit(f(x),x=c) " "6#-%&LimitG6$-%\"fG6#%\"xG/F)%\"cG" }{TEXT -1 8 " exists." }}{PARA 0 "" 0 "" {TEXT -1 36 "Then, if the value of this limit is " }{TEXT 302 1 "L" }{TEXT -1 31 ", the function f*, defined by: " }}{PARA 260 " " 0 "" {TEXT -1 2 " f" }{XPPEDIT 18 0 "`*`(x)=PIECEWISE([f(x),x<>c],[L ,x=c])" "6#/-%\"*G6#%\"xG-%*PIECEWISEG6$7$-%\"fG6#F'0F'%\"cG7$%\"LG/F' F0" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 17 "is continuous at " }{XPPEDIT 18 0 "x=c" "6#/%\"xG%\"cG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{PARA 0 "" 0 "" {TEXT -1 34 "Let f be the function defined by " }{XPPEDIT 18 0 " f(x)=1/x" "6#/-%\"fG6#%\"xG*&\"\"\"F)F'!\"\"" }{TEXT -1 34 ". f is co ntinuous on the interval" }{XPPEDIT 18 0 "``(0,infinity)" "6#-%!G6$\" \"!%)infinityG" }{TEXT -1 36 " and also continuous on the interval" } {XPPEDIT 18 0 "``(-infinity,0)" "6#-%!G6$,$%)infinityG!\"\"\"\"!" } {TEXT -1 42 ", but when considering the whole real line" }{XPPEDIT 18 0 "``(-infinity,infinity)" "6#-%!G6$,$%)infinityG!\"\"F'" }{TEXT -1 40 " we must say that f is discontinuous at " }{XPPEDIT 18 0 "x =0" "6 #/%\"xG\"\"!" }{TEXT -1 40 " simply because f is not defined there. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "plot(1/x,x=-4..4,y=-4..4,discont=true,thickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 287 309 309 {PLOTDATA 2 "6%-%'CURVESG6&7gn7$$!\"%\" \"!$!3++++++++D!#=7$$!3_5x]$Q6G\"R!#<$!3KI!yC92db#F-7$$!3!)oF\\M!\\p$Q F1$!3&os*H!QPig#F-7$$!3al+$)))Qj^PF1$!3E\"pyhH0bm#F-7$$!3#p#=+>KvlOF1$ !3#3Md/K_zs#F-7$$!310h+D2G!e$F1$!3WQ$)yqq2$z#F-7$$!3xf7$=yO5]$F1$!3A-F EbnHcGF-7$$!3JJ?moU)*=MF1$!3!*=t#)z_%[#HF-7$$!3b#3l@WDTL$F1$!3i1!>9F(G **HF-7$$!3#G#4+f(Q&\\KF1$!3%*zUx;)ft2$F-7$$!3e/;Me4`iJF1$!3c\"p#G#QC?; $F-7$$!3CW:;SW*e3$F1$!3nu*pja^0C$F-7$$!3_g2+s)>'**HF1$!3qJ7I^dvLLF-7$$ !3)y,u@0\"*H\"HF1$!3h$z/Gg(*GV$F-7$$!3k$)4MK\"3&HGF1$!3:w]oiJ=MNF-7$$! 39QfKVwp`FF1$!3RpA:c2[JOF-7$$!3]j&R$R;bjEF1$!3qWz=IcQaPF-7$$!3_%H#\\\\ h=(e#F1$!3uB^()GJ?lQF-7$$!3Q!H.0$\\N)\\#F1$!3+FvLWQj-SF-7$$!3X=s#)\\Us >CF1$!3hhDJ6BqKTF-7$$!3%>4L3$RXLBF1$!3-J.b\"z#\\&G%F-7$$!3!>R(*f=/8D#F 1$!3=LX&pwp=W%F-7$$!3k([N`a*el@F1$!3ofPC;-o+#F1$!3ypV+i*o]*\\F-7$$!31kd+XYy8>F1$!3fH[R>$[_ A&F-7$$!3cpf#oB:q$=F1$!3(QLEn)HhVaF-7$$!3#*G^#G<-Tv\"F1$!37Ifm*=A4q&F- 7$$!3[5Jmk[Wo;F1$!3#H&z#)4`g$*fF-7$$!3C32$)**ek%e\"F1$!3w338)3e0J'F-7$ $!3xnG\\#4mN]\"F1$!39(*oPd\\&3l'F-7$$!3X=H+\"F1$!3+%e7&4Qd!)**F-7$$!3cXTkOs#z<*F-$!3gmR@M1d*3\"F17$$!3 MEj&*H7\"F17$$!3yhB,S>#Q\\(F-$!3*f$yegDVM8F17$$!3'R& [++-CkmF-$!3y*\\#y`ha+:F17$$!3v(e,<0te\"eF-$!3J^z\"QIK%>PN)R-,#F17$$!3\\^rfrYU,UF-$!3q!=cy,X,Q#F17$$!3*4&f.!zd`J$F -$!3#o\"*HWqli,$F17$$!3ONwG3!\\G_#F-$!3w9D)**osP'RF17$$!3KjUk^$pyn\"F- $!3yO`Z/'R*ffF17$$!3U;IXxE\\t7F-$!3K9#yW\\?C&yF17$$!3A&p$!3a ttu#R%f]6!#;7$$!3IrKYC-P=lFgz$!3bcc#*4e7M:Fjz7$$!3SZ)3jT!eXVFgz$!3rtfn t&)=,BFjz7$$!3WN;B7b=fKFgz$!3WRYGV7DoIFjz7$$!3%QUa\"31zs@Fgz$!3J'3x+Iw Bg%Fjz7$$!3'y\"e6cJfH;Fgz$!3rPJI!)4]OhFjz7$$!3)=@xSq&R'3\"Fgz$!3+/m06# \\Z?*Fjz7$$!3#*)3z0ypz9)!#?$!3*[haNf*HF7!#:7$$!3.fgQ?D)>V&F\\]l$!3g!zt l[[4%=F_]l7$$!3eHI>g_*fr#F\\]l$!3/R%41b\"*=o$F_]l7$$!3]+++++++!)!#E$!3 ')************\\7!\"*7gn7$$\"3]+++++++!)F]^l$\"3')************\\7F`^l7 $$\"3KWSQlAlCFF\\]l$\"3[7'Q/7$>qOF_]l7$$\"3_)3o2`'H\\aF\\]l$\"3^pnGa#* 4N=F_]l7$$\"39L@:'zSR<)F\\]l$\"3qw1a,,SB7F_]l7$$\"3y^7F-$\"39t950-M#*zF17$$\"3% QJs]t40j\"F-$\"3*eR#y\"R^I8'F17$$\"3CW$*p\">hO[#F-$\"3*4*e\\pSJESF17$$ \"3(>%F-$\"3;k!R:\"[a#Q#F 17$$\"3q,uohAj*)\\F-$\"3edFR7d:/?F17$$\"3i%ozLRd,\"eF-$\"3L\\T[qP7@\"F17$$\"3NbXQyc0T\"*F-$\"3=w`p\\b' R4\"F17$$\"3pR#**f8!Q+5F1$\"3e\"G#\\%3+i***F-7$$\"37#)f#e&*3q3\"F1$\"3 1%e'H>ib*>*F-7$$\"3O;!fc(=\\q6F1$\"3*4%HkQoTV&)F-7$$\"3'=1uYO-jC\"F1$ \"3^(Q8q2NP-)F-7$$\"3[O/mo$[kL\"F1$\"3Mi\"ysw=D[(F-7$$\"3Z0x]eQ\"GT\"F 1$\"31X+xkK2yqF-7$$\"3$)4n\\x]k,:F1$\"3=L&[4Aj$fmF-7$$\"3a\"ys\"edF!e \"F1$\"37'peM\\4!GjF-7$$\"3%y!p;xgam;F1$\"3D_F1$ \"3f'R(oPZ0F_F-7$$\"39:1nXc-)*>F1$\"3!35KKwS\\+&F-7$$\"3;OU*HO:i3#F1$ \"3)[(y%o(*oLz%F-7$$\"3kISHp\"3TN:CF1 $\"3&f:g0oz,9%F-7$$\"3WKr]:RV'\\#F1$\"3S=#oc)Qr0SF-7$$\"3w\"3Fy@fke#F1 $\"3)=mE^X*GmQF-7$$\"3EJ')*f&4NnEF1$\"3C%G$*\\'z.\\PF-7$$\"3')fD\\_,s` FF1$\"3G/#)3s5XJOF-7$$\"3j*p-:[$)>$GF1$\"30;E]US4JNF-7$$\"3=3\\;sfa " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "Note that there is a \+ large difference between the value of the function at a small negativ e number such as " }{XPPEDIT 18 0 "-10^(-6);" "6#,$)\"#5,$\"\"'!\"\"F( " }{TEXT -1 43 " and the value at the small positive number" } {XPPEDIT 18 0 "10^(-6);" "6#)\"#5,$\"\"'!\"\"" }{TEXT -1 19 ", the val ues being " }{XPPEDIT 18 0 "-10^6" "6#,$*$\"#5\"\"'!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "10^6" "6#*$\"#5\"\"'" }{TEXT -1 15 " respectiv ely. " }}{PARA 0 "" 0 "" {TEXT -1 21 "The discontinuity at " } {XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 4 " is " }{TEXT 261 12 " nonremovable" }{TEXT -1 37 " because there is no way of defining " } {XPPEDIT 18 0 "f(0)" "6#-%\"fG6#\"\"!" }{TEXT -1 45 " so as to make th e function continuous at 0. " }}{PARA 0 "" 0 "" {TEXT -1 38 "For examp le the function F defined by " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "F(x)=PIECEWISE([1/x,x<>0],[0,x=0])" "6#/-%\"FG6#%\"xG-% *PIECEWISEG6$7$*&\"\"\"F-F'!\"\"0F'\"\"!7$F0/F'F0" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 20 "is discintinuous at " }{XPPEDIT 18 0 "x=0 " "6#/%\"xG\"\"!" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 192 "p1 := plot(1/x,x=-4..4,y=-4..4,discont=true,thicknes s=2):\np2 := plot([[[0,0]]$4],style=point,symbol=[circle$2,diamond,cro ss],\n symbolsize=[15,10$3],color=red):\nplots[display]([p1,p2 ]);" }}{PARA 13 "" 1 "" {GLPLOT2D 293 296 296 {PLOTDATA 2 "6)-%'CURVES G6&7gn7$$!\"%\"\"!$!3++++++++D!#=7$$!3_5x]$Q6G\"R!#<$!3KI!yC92db#F-7$$ !3!)oF\\M!\\p$QF1$!3&os*H!QPig#F-7$$!3al+$)))Qj^PF1$!3E\"pyhH0bm#F-7$$ !3#p#=+>KvlOF1$!3#3Md/K_zs#F-7$$!310h+D2G!e$F1$!3WQ$)yqq2$z#F-7$$!3xf7 $=yO5]$F1$!3A-FEbnHcGF-7$$!3JJ?moU)*=MF1$!3!*=t#)z_%[#HF-7$$!3b#3l@WDT L$F1$!3i1!>9F(G**HF-7$$!3#G#4+f(Q&\\KF1$!3%*zUx;)ft2$F-7$$!3e/;Me4`iJF 1$!3c\"p#G#QC?;$F-7$$!3CW:;SW*e3$F1$!3nu*pja^0C$F-7$$!3_g2+s)>'**HF1$! 3qJ7I^dvLLF-7$$!3)y,u@0\"*H\"HF1$!3h$z/Gg(*GV$F-7$$!3k$)4MK\"3&HGF1$!3 :w]oiJ=MNF-7$$!39QfKVwp`FF1$!3RpA:c2[JOF-7$$!3]j&R$R;bjEF1$!3qWz=IcQaP F-7$$!3_%H#\\\\h=(e#F1$!3uB^()GJ?lQF-7$$!3Q!H.0$\\N)\\#F1$!3+FvLWQj-SF -7$$!3X=s#)\\Us>CF1$!3hhDJ6BqKTF-7$$!3%>4L3$RXLBF1$!3-J.b\"z#\\&G%F-7$ $!3!>R(*f=/8D#F1$!3=LX&pwp=W%F-7$$!3k([N`a*el@F1$!3ofPC;-o+#F1$!3ypV+i*o]*\\F-7$$!31kd+XYy 8>F1$!3fH[R>$[_A&F-7$$!3cpf#oB:q$=F1$!3(QLEn)HhVaF-7$$!3#*G^#G<-Tv\"F1 $!37Ifm*=A4q&F-7$$!3[5Jmk[Wo;F1$!3#H&z#)4`g$*fF-7$$!3C32$)**ek%e\"F1$! 3w338)3e0J'F-7$$!3xnG\\#4mN]\"F1$!39(*oPd\\&3l'F-7$$!3X=H+\"F1$!3+%e7&4Qd!)**F-7$$!3cXTkOs#z<*F-$!3gmR @M1d*3\"F17$$!3MEj&*H7\"F17$$!3yhB,S>#Q\\(F-$!3*f$ye gDVM8F17$$!3'R&[++-CkmF-$!3y*\\#y`ha+:F17$$!3v(e,<0te\"eF-$!3J^z\"QIK% >PN)R-,#F17$$!3\\^rfrYU,UF-$!3q!=cy,X,Q#F17$ $!3*4&f.!zd`J$F-$!3#o\"*HWqli,$F17$$!3ONwG3!\\G_#F-$!3w9D)**osP'RF17$$ !3KjUk^$pyn\"F-$!3yO`Z/'R*ffF17$$!3U;IXxE\\t7F-$!3K9#yW\\?C&yF17$$!3A& p$!3attu#R%f]6!#;7$$!3IrKYC-P=lFgz$!3bcc#*4e7M:Fjz7$$!3SZ)3j T!eXVFgz$!3rtfnt&)=,BFjz7$$!3WN;B7b=fKFgz$!3WRYGV7DoIFjz7$$!3%QUa\"31z s@Fgz$!3J'3x+IwBg%Fjz7$$!3'y\"e6cJfH;Fgz$!3rPJI!)4]OhFjz7$$!3)=@xSq&R' 3\"Fgz$!3+/m06#\\Z?*Fjz7$$!3#*)3z0ypz9)!#?$!3*[haNf*HF7!#:7$$!3.fgQ?D) >V&F\\]l$!3g!ztl[[4%=F_]l7$$!3eHI>g_*fr#F\\]l$!3/R%41b\"*=o$F_]l7$$!3] +++++++!)!#E$!3')************\\7!\"*7gn7$$\"3]+++++++!)F]^l$\"3')***** *******\\7F`^l7$$\"3KWSQlAlCFF\\]l$\"3[7'Q/7$>qOF_]l7$$\"3_)3o2`'H\\aF \\]l$\"3^pnGa#*4N=F_]l7$$\"39L@:'zSR<)F\\]l$\"3qw1a,,SB7F_]l7$$\"3y^7F-$\"39t9 50-M#*zF17$$\"3%QJs]t40j\"F-$\"3*eR#y\"R^I8'F17$$\"3CW$*p\">hO[#F-$\"3 *4*e\\pSJESF17$$\"3(>%F-$ \"3;k!R:\"[a#Q#F17$$\"3q,uohAj*)\\F-$\"3edFR7d:/?F17$$\"3i%ozLRd,\"eF- $\"3L\\T[qP7@\"F17$$\"3NbXQyc0T \"*F-$\"3=w`p\\b'R4\"F17$$\"3pR#**f8!Q+5F1$\"3e\"G#\\%3+i***F-7$$\"37# )f#e&*3q3\"F1$\"31%e'H>ib*>*F-7$$\"3O;!fc(=\\q6F1$\"3*4%HkQoTV&)F-7$$ \"3'=1uYO-jC\"F1$\"3^(Q8q2NP-)F-7$$\"3[O/mo$[kL\"F1$\"3Mi\"ysw=D[(F-7$ $\"3Z0x]eQ\"GT\"F1$\"31X+xkK2yqF-7$$\"3$)4n\\x]k,:F1$\"3=L&[4Aj$fmF-7$ $\"3a\"ys\"edF!e\"F1$\"37'peM\\4!GjF-7$$\"3%y!p;xgam;F1$\"3D_F1$\"3f'R(oPZ0F_F-7$$\"39:1nXc-)*>F1$\"3!35KKwS\\+&F-7$$ \"3;OU*HO:i3#F1$\"3)[(y%o(*oLz%F-7$$\"3kISHp\"3TN:CF1$\"3&f:g0oz,9%F-7$$\"3WKr]:RV'\\#F1$\"3S=#oc)Qr0SF-7$$ \"3w\"3Fy@fke#F1$\"3)=mE^X*GmQF-7$$\"3EJ')*f&4NnEF1$\"3C%G$*\\'z.\\PF- 7$$\"3')fD\\_,s`FF1$\"3G/#)3s5XJOF-7$$\"3j*p-:[$)>$GF1$\"30;E]US4JNF-7 $$\"3=3\\;sfa " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "Because F is defined at " }{XPPEDIT 18 0 "x=0" "6#/%\" xG\"\"!" }{TEXT -1 63 ", the technical reason for the fact that F is d iscontinuous at " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 9 " i s that " }{XPPEDIT 18 0 "Limit(F(x),x=0)" "6#-%&LimitG6$-%\"FG6#%\"xG/ F)\"\"!" }{TEXT -1 17 " does not exist. " }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "Limit(F(x),x = 0)" "6#-%&LimitG6$-%\"F G6#%\"xG/F)\"\"!" }{TEXT -1 24 " does not exist because " }{XPPEDIT 18 0 "Limit(F(x),x = 0^`+`)=Limit(1/x,x = 0^`+`)" "6#/-%&LimitG6$-%\"F G6#%\"xG/F*)\"\"!%\"+G-F%6$*&\"\"\"F2F*!\"\"/F*)F-F." }{XPPEDIT 18 0 " ``=infinity" "6#/%!G%)infinityG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "L imit(F(x),x = 0^`-`)=Limit(1/x,x = 0^`-`)" "6#/-%&LimitG6$-%\"FG6#%\"x G/F*)\"\"!%\"-G-F%6$*&\"\"\"F2F*!\"\"/F*)F-F." }{XPPEDIT 18 0 "``=-inf inity" "6#/%!G,$%)infinityG!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Ex ample 2 " }}{PARA 0 "" 0 "" {TEXT -1 33 "Let g be the function defined by " }{XPPEDIT 18 0 "g(x)=(x^2-1)/(x-1)" "6#/-%\"gG6#%\"xG*&,&*$F'\" \"#\"\"\"F,!\"\"F,,&F'F,F,F-F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 41 "g is defined for all real numbers except " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 33 ". g is continuous on the interv al" }{XPPEDIT 18 0 "``(-infinity,1)" "6#-%!G6$,$%)infinityG!\"\"\"\"\" " }{TEXT -1 36 " and also continuous on the interval" }{XPPEDIT 18 0 " ``(1,infinity)" "6#-%!G6$\"\"\"%)infinityG" }{TEXT -1 26 ", but is dis continuous at " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 9 " be cause " }{XPPEDIT 18 0 "g(1)" "6#-%\"gG6#\"\"\"" }{TEXT -1 17 " does n ot exist. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 226 "p1 := plot(x+1,x=-2..0.96,thickness=2):\np2 := plo t(x+1,x=1.04..3,thickness=2):\np3 := plot([[[1,2]]$2],style=point,colo r=red,symbol=[circle$2],symbolsize=[15,18]):\nplots[display]([p1,p2,p3 ],labels=[`x`,`y`],title=\"graph of g\");" }}{PARA 13 "" 1 "" {GLPLOT2D 273 256 256 {PLOTDATA 2 "6)-%'CURVESG6%7S7$$!\"#\"\"!$!\"\"F *7$$!3tmmmB/[N>!#<$!3GnmmOU![N*!#=7$$!3CLLG&GU$z=F0$!3YKL$G&GU$z)F37$$ !3fmmOx!4i\"=F0$!3!fmmOx!4i\")F37$$!3immc\"QdEv\"F0$!3\"F0$!3A******pP8c>F37$$!33++?;g$Q8\"F0$!3u+++i,OQ8F37$$!3 kmm@aitx5F0$!3Qjmm@aitx!#>7$$!3ILL86#G5,\"F0$!3))HLL86#G5\"Fdp7$$!3yLL L&[x^a*F3$\"3MimmY^A[XFdp7$$!3G****\\jk#y)))F3$\"3t++]ON<76F37$$!3`LLL D%ffI)F3$\"3Ymmmu0/%p\"F37$$!3%z***\\j!fvm(F3$\"31-+]O4WKBF37$$!3C)*** \\]4lfqF3$\"3v,+]\\!\\.%HF37$$!3^JLL3EODkF3$\"3]omm\"RPYd$F37$$!3qJL$o !z)G%eF3$\"3Iom;$47r:%F37$$!3.nmm^-h9_F3$\"3)HLL$[(*Q&y%F37$$!3ymm;Uj+ iXF3$\"3ALL$yl$*zV&F37$$!37****\\?F\"R*RF3$\"3)3++&zs31gF37$$!3#fmmc/c .Q$F3$\"33MLLaRk>mF37$$!3i*****R'>\\YFF3$\"3Q+++O!3ND(F37$$!3%4++Iizj7 #F3$\"31*****pP?O(yF37$$!3g)***\\Z!*QE:F3$\"3S,+]_4ht%)F37$$!3h2++!4=? g)Fdp$\"3C*****4>)zR\"*F37$$!38YLL`%Hgh#Fdp$\"3RlmmaqRQ(*F37$$\"376+++ 6HvPFdp$\"36+++6HvP5F07$$\"3%)>LLeuxm&*Fdp$\"3?LLeuxm&4\"F07$$\"3w.++y ,%)*e\"F3$\"3Q++!y,%)*e6F07$$\"3+km;$o'f&=#F3$\"3SmmJo'f&=7F07$$\"3/,+ ]!*QL3GF3$\"35++0*QL3G\"F07$$\"3*ymmmLtrT$F3$\"3zmmmLtrT8F07$$\"3w++]7 =daSF3$\"33++D\"=daS\"F07$$\"3'fLL8Ii%oYF3$\"3gLL8Ii%oY\"F07$$\"3#HLLB *RD'H&F3$\"3HLLB*RD'H:F07$$\"35qm;^s%)=fF3$\"3+nm6DZ)=f\"F07$$\"3w-++' zX4\\'F3$\"3G++gzX4\\;F07$$\"3Snmm*GNm9(F3$\"3umm'*GNm9F0-%'COLOURG6&%$RGBG$\"#5F,$F*F*F`[l-%*THICKNESSG6#\"\"#-F$6%7S7 $$\"3/++++++S5F0$\"3/++++++S?F07$$\"3qmm;ACs#3\"F0$\"3[mm;ACs#3#F07$$ \"3VL$eK(\\*)>6F0$\"3VL$eK(\\*)>@F07$$\"3smmr%R*ph6F0$\"3%pm;ZR*ph@F07 $$\"3tmm,B4y.7F0$\"3tmm,B4y.AF07$$\"3OL$e^WicC\"F0$\"3eL$e^WicC#F07$$ \"3wm;Rx>\\%G\"F0$\"3wm;Rx>\\%G#F07$$\"30+]#*3xpC8F0$\"30+]#*3xpCBF07$ $\"3wm;*R`yiO\"F0$\"3wm;*R`yiO#F07$$\"35+]#)3gs29F0$\"35+]#)3gs2CF07$$ \"3YLLBJ)f.X\"F0$\"3YLLBJ)f.X#F07$$\"3qmm@D<\"z[\"F0$\"3qmm@D<\"z[#F07 $$\"35++qji=I:F0$\"35++qji=IDF07$$\"3+++]&QMEd\"F0$\"3y****\\&QMEd#F07 $$\"38++I;5a8;F0$\"38++I;5a8EF07$$\"3'om\"*f:)o];F0$\"3'om\"*f:)o]EF07 $$\"3YLL.)pf[p\"F0$\"3YLL.)pf[p#F07$$\"3DLL8)yyAt\"F0$\"3DLL8)yyAt#F07 $$\"3'***\\_&31ex\"F0$\"3u**\\_&31ex#F07$$\"3BLL8>^L9=F0$\"3BLL8>^L9GF 07$$\"3++]_vvgc=F0$\"3++]_vvgcGF07$$\"3-+]d]4'o*=F0$\"3E+]d]4'o*GF07$$ \"3PLLeC6')Q>F0$\"3:LLeC6')QHF07$$\"3WL$e`RIu(>F0$\"3mL$e`RIu(HF07$$\" 3XmmTkD.>?F0$\"3XmmTkD.>IF07$$\"3+n;*f_XA1#F0$\"3+n;*f_XA1$F07$$\"3G+] 2OD')*4#F0$\"3G+]2OD')*4$F07$$\"3hmm^P**[S@F0$\"3hmm^P**[SJF07$$\"3))* ***f=?Y#=#F0$\"3))****f=?Y#=$F07$$\"3#)***\\9^BNA#F0$\"3#)***\\9^BNA$F 07$$\"3/+]7+ZBF0$\"33LL8n>+ZLF07$$\"3G++]tGK*Q#F0$\"3G++]tGK*Q$F07$$\"3hm; z/>nFCF0$\"3hm;z/>nFMF07$$\"3A++IDvfpCF0$\"3A++IDvfpMF07$$\"3KL$eoEY!4 DF0$\"3KL$eoEY!4NF07$$\"3)***\\Uo:G]DF0$\"3)***\\Uo:G]NF07$$\"33LL$Gh' f!f#F0$\"33LL$Gh'f!f$F07$$\"3;+]7GF!Gj#F0$\"3;+]7GF!Gj$F07$$\"3gmmYLAX tEF0$\"3gmmYLAXtOF07$$\"3!om;QAA]r#F0$\"3!om;QAA]r$F07$$\"3]L$eO*zCcFF 0$\"3]L$eO*zCcPF07$$\"37++g%>ITz#F0$\"37++g%>ITz$F07$$\"36LL)=ZZv$GF0$ \"36LL)=ZZv$QF07$$\"3OmmE-/QwGF0$\"3OmmE-/QwQF07$$\"3'***\\_0WyLTdHF0$\"35+]n>LTdRF07$$\"\"$F*$\"\"%F*FjzFa [l-F$6&7#7$$\"\"\"F*$Fd[lF*-%'SYMBOLG6$%'CIRCLEG\"#:-F[[l6&F][l$\"*+++ +\"!\")F`[lF`[l-%&STYLEG6#%&POINTG-F$6&F_[m-Fe[m6$Fg[m\"#=Fi[mF^\\m-%+ AXESLABELSG6%%\"xG%\"yG-%%FONTG6#%(DEFAULTG-%&TITLEG6#Q+graph~of~g6\"- %%VIEWG6$;F(FijlF_]m" 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" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "The discontinuity at " }{XPPEDIT 18 0 " x=1" "6#/%\"xG\"\"\"" }{TEXT -1 4 " is " }{TEXT 261 9 "removable" } {TEXT -1 121 " since we can obtain a continuous function g* by letting g* have the same value as g where g is defined and then letting " } {XPPEDIT 18 0 "g*`*`(1) = 2;" "6#/*&%\"gG\"\"\"-%\"*G6#F&F&\"\"#" } {TEXT -1 13 ". , that is, " }}{PARA 257 "" 0 "" {TEXT -1 2 " g" } {XPPEDIT 18 0 "`*`(x) = PIECEWISE([g(x), x <> 1],[2, x = 1]);" "6#/-% \"*G6#%\"xG-%*PIECEWISEG6$7$-%\"gG6#F'0F'\"\"\"7$\"\"#/F'F0" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 257 "" 0 "" {TEXT -1 2 " g" }{XPPEDIT 18 0 "`*`(x) = PIECEWISE([(x^2-1)/(x-1), x < > 1],[2, x = 1]);" "6#/-%\"*G6#%\"xG-%*PIECEWISEG6$7$*&,&*$F'\"\"#\"\" \"F0!\"\"F0,&F'F0F0F1F10F'F07$F//F'F0" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 257 "" 0 "" {TEXT -1 2 " g" }{XPPEDIT 18 0 "`*`(x) = PIECEWISE([x+1, x <> 1],[2, x = 1]);" "6#/-%\"*G6#%\"xG-%* PIECEWISEG6$7$,&F'\"\"\"F-F-0F'F-7$\"\"#/F'F-" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 257 "" 0 "" {TEXT -1 2 " g" } {XPPEDIT 18 0 "`*`(x) = x+1;" "6#/-%\"*G6#%\"xG,&F'\"\"\"F)F)" }{TEXT -1 22 " for all real numbers " }{TEXT 291 1 "x" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "plot(x+1,x=-2..3,thickness=2,labels=[`x`,`y`],title=\"graph of g* \");" }}{PARA 13 "" 1 "" {GLPLOT2D 291 273 273 {PLOTDATA 2 "6'-%'CURVE SG6$7S7$$!\"#\"\"!$!\"\"F*7$$!3smm;HU,\"*=!#<$!3Fnmm\"HU,\"*)!#=7$$!3S L$3FH'='z\"F0$!35ML3FH'='zF37$$!3gmmTgBa*o\"F0$!35mm;/OU&*oF37$$!3amm \"H_\">#e\"F0$!3Plm;H_\">#eF37$$!3ML$3_!4Nv9F0$!3SLL3_!4Nv%F37$$!3km;/ wfHw8F0$!3YmmTg(fHw$F37$$!3;+]PM.tt7F0$!3i,+vVLIPFF37$$!3em;/,oln6F0$! 3#em;/,oln\"F37$$!3%)**\\(oWB>1\"F0$!3O%)**\\(oWB>'!#>7$$!3eJLLepjJ&*F 3$\"3C%ommTIOo%Ffn7$$!3Ulm;z/ot&)F3$\"3eML$3_>jU\"F37$$!3u)****\\P[_\\ (F3$\"3E,++D;v/DF37$$!3A*****\\7)Q7kF3$\"3y+++v=h(e$F37$$!3e*****\\i^) o`F3$\"3V+++v$[6j%F37$$!3vlmT50A@WF3$\"3EMLe*[z(ybF37$$!3OKLLeaR%H$F3$ \"3knmmTXg0nF37$$!3kJLLLo#)RBF3$\"3OommmJpxgF3$\"3fL$e*)>pxg\"F07$$\"3w++v$f4t. (F3$\"33+]Pf4t.F07$$\"3:++DJE>>5F0$\"3:++DJE>>?F07$$\"3F+]i!RU07\"F0$\" 3F+]i!RU07#F07$$\"3+++v=S2L7F0$\"3+++v=S2LAF07$$\"3Jmmm\"p)=M8F0$\"3Jm mm\"p)=MBF07$$\"3B++](=]@W\"F0$\"3B++](=]@W#F07$$\"35L$e*[$z*R:F0$\"35 L$e*[$z*RDF07$$\"3e++]iC$pk\"F0$\"3e++]iC$pk#F07$$\"3[m;H2qcZF0$\"3Ymm; /OgbHF07$$\"3w**\\ilAFj?F0$\"3w**\\ilAFjIF07$$\"3yLLL$)*pp;#F0$\"3yLLL $)*pp;$F07$$\"3)RL$3xe,tAF0$\"3)RL$3xe,tKF07$$\"3Cn;HdO=yBF0$\"3Cn;HdO =yLF07$$\"3a+++D>#[Z#F0$\"3a+++D>#[Z$F07$$\"3SnmT&G!e&e#F0$\"3SnmT&G!e &e$F07$$\"3#RLLL)Qk%o#F0$\"3#RLLL)Qk%o$F07$$\"37+]iSjE!z#F0$\"37+]iSjE !z$F07$$\"3a+]P40O\"*GF0$\"3a+]P40O\"*QF07$$\"\"$F*$\"\"%F*-%'COLOURG6 &%$RGBG$\"#5F,$F*F*F`[l-%+AXESLABELSG6$%\"xG%\"yG-%&TITLEG6#Q,graph~of ~g*6\"-%*THICKNESSG6#\"\"#-%%VIEWG6$;F(Ffz%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 4 "Note" }{TEXT -1 42 ": If we chose a value diffe rent from 2 at " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 95 ", then we obtain a function that is discontinuous at 1. For example, th e function G defined by " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "G(x)=PIECEWISE([x+1, x <> 1],[3, x = 1])" "6#/-%\"GG6#% \"xG-%*PIECEWISEG6$7$,&F'\"\"\"F-F-0F'F-7$\"\"$/F'F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 67 "is discontinuous at 1. The technical re ason for this is that while " }{XPPEDIT 18 0 "G(1)" "6#-%\"GG6#\"\"\" " }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Limit(G(x),x=1)" "6#-%&LimitG6$- %\"GG6#%\"xG/F)\"\"\"" }{TEXT -1 60 " both exist, they have different \+ values, since G(1)=3 while " }{XPPEDIT 18 0 "Limit(G(x),x=1)=2" "6#/-% &LimitG6$-%\"GG6#%\"xG/F*\"\"\"\"\"#" }{TEXT -1 52 ". The third condit ion in the test for continuity at " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\" \"" }{TEXT -1 8 " fails. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 333 "p1 := plot(x+1,x=-2..0.96,thicknes s=2):\np2 := plot(x+1,x=1.04..3,thickness=2):\np3 := plot([[[1,2]]$2], style=point,color=red,symbol=[circle$2],symbolsize=[15,18]):\np4 := pl ot([[[1,3]]$4],style=point,color=red,symbol=[circle$2,diamond,cross], \n symbolsize=[15,10$3]):\nplots[display]([p||(1..4)],labels=[`x`, `y`],title=\"graph of G\");" }}{PARA 13 "" 1 "" {GLPLOT2D 292 279 279 {PLOTDATA 2 "6--%'CURVESG6%7S7$$!\"#\"\"!$!\"\"F*7$$!3tmmmB/[N>!#<$!3G nmmOU![N*!#=7$$!3CLLG&GU$z=F0$!3YKL$G&GU$z)F37$$!3fmmOx!4i\"=F0$!3!fmm Ox!4i\")F37$$!3immc\"QdEv\"F0$!3\"F0$!3A** ****pP8c>F37$$!33++?;g$Q8\"F0$!3u+++i,OQ8F37$$!3kmm@aitx5F0$!3Qjmm@ait x!#>7$$!3ILL86#G5,\"F0$!3))HLL86#G5\"Fdp7$$!3yLLL&[x^a*F3$\"3MimmY^A[X Fdp7$$!3G****\\jk#y)))F3$\"3t++]ON<76F37$$!3`LLLD%ffI)F3$\"3Ymmmu0/%p \"F37$$!3%z***\\j!fvm(F3$\"31-+]O4WKBF37$$!3C)***\\]4lfqF3$\"3v,+]\\! \\.%HF37$$!3^JLL3EODkF3$\"3]omm\"RPYd$F37$$!3qJL$o!z)G%eF3$\"3Iom;$47r :%F37$$!3.nmm^-h9_F3$\"3)HLL$[(*Q&y%F37$$!3ymm;Uj+iXF3$\"3ALL$yl$*zV&F 37$$!37****\\?F\"R*RF3$\"3)3++&zs31gF37$$!3#fmmc/c.Q$F3$\"33MLLaRk>mF3 7$$!3i*****R'>\\YFF3$\"3Q+++O!3ND(F37$$!3%4++Iizj7#F3$\"31*****pP?O(yF 37$$!3g)***\\Z!*QE:F3$\"3S,+]_4ht%)F37$$!3h2++!4=?g)Fdp$\"3C*****4>)zR \"*F37$$!38YLL`%Hgh#Fdp$\"3RlmmaqRQ(*F37$$\"376+++6HvPFdp$\"36+++6HvP5 F07$$\"3%)>LLeuxm&*Fdp$\"3?LLeuxm&4\"F07$$\"3w.++y,%)*e\"F3$\"3Q++!y,% )*e6F07$$\"3+km;$o'f&=#F3$\"3SmmJo'f&=7F07$$\"3/,+]!*QL3GF3$\"35++0*QL 3G\"F07$$\"3*ymmmLtrT$F3$\"3zmmmLtrT8F07$$\"3w++]7=daSF3$\"33++D\"=daS \"F07$$\"3'fLL8Ii%oYF3$\"3gLL8Ii%oY\"F07$$\"3#HLLB*RD'H&F3$\"3HLLB*RD' H:F07$$\"35qm;^s%)=fF3$\"3+nm6DZ)=f\"F07$$\"3w-++'zX4\\'F3$\"3G++gzX4 \\;F07$$\"3Snmm*GNm9(F3$\"3umm'*GNm9F0-%'COLOUR G6&%$RGBG$\"#5F,$F*F*F`[l-%*THICKNESSG6#\"\"#-F$6%7S7$$\"3/++++++S5F0$ \"3/++++++S?F07$$\"3qmm;ACs#3\"F0$\"3[mm;ACs#3#F07$$\"3VL$eK(\\*)>6F0$ \"3VL$eK(\\*)>@F07$$\"3smmr%R*ph6F0$\"3%pm;ZR*ph@F07$$\"3tmm,B4y.7F0$ \"3tmm,B4y.AF07$$\"3OL$e^WicC\"F0$\"3eL$e^WicC#F07$$\"3wm;Rx>\\%G\"F0$ \"3wm;Rx>\\%G#F07$$\"30+]#*3xpC8F0$\"30+]#*3xpCBF07$$\"3wm;*R`yiO\"F0$ \"3wm;*R`yiO#F07$$\"35+]#)3gs29F0$\"35+]#)3gs2CF07$$\"3YLLBJ)f.X\"F0$ \"3YLLBJ)f.X#F07$$\"3qmm@D<\"z[\"F0$\"3qmm@D<\"z[#F07$$\"35++qji=I:F0$ \"35++qji=IDF07$$\"3+++]&QMEd\"F0$\"3y****\\&QMEd#F07$$\"38++I;5a8;F0$ \"38++I;5a8EF07$$\"3'om\"*f:)o];F0$\"3'om\"*f:)o]EF07$$\"3YLL.)pf[p\"F 0$\"3YLL.)pf[p#F07$$\"3DLL8)yyAt\"F0$\"3DLL8)yyAt#F07$$\"3'***\\_&31ex \"F0$\"3u**\\_&31ex#F07$$\"3BLL8>^L9=F0$\"3BLL8>^L9GF07$$\"3++]_vvgc=F 0$\"3++]_vvgcGF07$$\"3-+]d]4'o*=F0$\"3E+]d]4'o*GF07$$\"3PLLeC6')Q>F0$ \"3:LLeC6')QHF07$$\"3WL$e`RIu(>F0$\"3mL$e`RIu(HF07$$\"3XmmTkD.>?F0$\"3 XmmTkD.>IF07$$\"3+n;*f_XA1#F0$\"3+n;*f_XA1$F07$$\"3G+]2OD')*4#F0$\"3G+ ]2OD')*4$F07$$\"3hmm^P**[S@F0$\"3hmm^P**[SJF07$$\"3))****f=?Y#=#F0$\"3 ))****f=?Y#=$F07$$\"3#)***\\9^BNA#F0$\"3#)***\\9^BNA$F07$$\"3/+]7+ZBF0$\"3 3LL8n>+ZLF07$$\"3G++]tGK*Q#F0$\"3G++]tGK*Q$F07$$\"3hm;z/>nFCF0$\"3hm;z />nFMF07$$\"3A++IDvfpCF0$\"3A++IDvfpMF07$$\"3KL$eoEY!4DF0$\"3KL$eoEY!4 NF07$$\"3)***\\Uo:G]DF0$\"3)***\\Uo:G]NF07$$\"33LL$Gh'f!f#F0$\"33LL$Gh 'f!f$F07$$\"3;+]7GF!Gj#F0$\"3;+]7GF!Gj$F07$$\"3gmmYLAXtEF0$\"3gmmYLAXt OF07$$\"3!om;QAA]r#F0$\"3!om;QAA]r$F07$$\"3]L$eO*zCcFF0$\"3]L$eO*zCcPF 07$$\"37++g%>ITz#F0$\"37++g%>ITz$F07$$\"36LL)=ZZv$GF0$\"36LL)=ZZv$QF07 $$\"3OmmE-/QwGF0$\"3OmmE-/QwQF07$$\"3'***\\_0WyLTdHF0$\"35+]n>LTdRF07$$\"\"$F*$\"\"%F*FjzFa[l-F$6&7#7$$\"\" \"F*$Fd[lF*-%'SYMBOLG6$%'CIRCLEG\"#:-F[[l6&F][l$\"*++++\"!\")F`[lF`[l- %&STYLEG6#%&POINTG-F$6&F_[m-Fe[m6$Fg[m\"#=Fi[mF^\\m-F$6&7#7$Fa[mFijlFd [mFi[mF^\\m-F$6&Fi\\m-Fe[m6$Fg[mF_[lFi[mF^\\m-F$6&Fi\\m-Fe[m6$%(DIAMON DGF_[lFi[mF^\\m-F$6&Fi\\m-Fe[m6$%&CROSSGF_[lFi[mF^\\m-%+AXESLABELSG6%% \"xG%\"yG-%%FONTG6#%(DEFAULTG-%&TITLEG6#Q+graph~of~G6\"-%%VIEWG6$;F(Fi jlFa^m" 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" "Curve 7" "Curve 8" }}}}{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 33 "Let h be the function defined by " } {XPPEDIT 18 0 "h(x) = PIECEWISE([x+1, x <= 0],[x^2+1, 0 < x])" "6#/-% \"hG6#%\"xG-%*PIECEWISEG6$7$,&F'\"\"\"F-F-1F'\"\"!7$,&*$F'\"\"#F-F-F-2 F/F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 28 "h is continuous e verywhere. " }}{PARA 0 "" 0 "" {TEXT -1 33 "To check that h is continu ous at " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 11 " note that " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(h(x),x = 0 ^`-`) = Limit(``(x+1),x = 0^`-`);" "6#/-%&LimitG6$-%\"hG6#%\"xG/F*)\" \"!%\"-G-F%6$-%!G6#,&F*\"\"\"F5F5/F*)F-F." }{XPPEDIT 18 0 "``=1" "6#/% !G\"\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(h(x),x = 0^`+`) = Lim it(``(x^2+1),x = 0^`+`);" "6#/-%&LimitG6$-%\"hG6#%\"xG/F*)\"\"!%\"+G-F %6$-%!G6#,&*$F*\"\"#\"\"\"F7F7/F*)F-F." }{XPPEDIT 18 0 "``=1" "6#/%!G \"\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 7 "so that" }} {PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Limit(h(x),x=0)=1" " 6#/-%&LimitG6$-%\"hG6#%\"xG/F*\"\"!\"\"\"" }{XPPEDIT 18 0 "``=h(0)" "6 #/%!G-%\"hG6#\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "h := x -> piecewise(x<0,x+1 ,x>=0,x^2+1):'h(x)'=h(x);\nplot(h(x),x=-2..2,y=-0.8..4,thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"hG6#%\"xG-%*PIECEWISEG6$7$,&F' \"\"\"F-F-2F'\"\"!7$,&*$)F'\"\"#F-F-F-F-1F/F'" }}{PARA 13 "" 1 "" {GLPLOT2D 303 353 353 {PLOTDATA 2 "6&-%'CURVESG6$7U7$$!\"#\"\"!$!\"\"F *7$$!3MLLL$Q6G\">!#<$!3PLLLLQ6G\"*!#=7$$!3bmm;M!\\p$=F0$!3]lmmT.\\p$)F 37$$!3MLLL))Qj^7$$!3w++++()>'***F3$\"3HI#*******H,Q!#@7$$!3E+ +++0\"*H\"*F3$\"3Q(*******\\*3q)F`o7$$!35++++83&H)F3$\"3!********p=\\q \"F37$$!3\\LLL3k(p`(F3$\"3_mmm\"fBIY#F37$$!3Anmmmj^NmF3$\"3yKLLLO[kLF3 7$$!3)zmmmYh=(eF3$\"3.KLLL&Q\"GTF37$$!3+,++v#\\N)\\F3$\"3+*****\\s]k,& F37$$!3commmCC(>%F3$\"3WJLLLvv-eF37$$!39*****\\FRXL$F3$\"3'3++]sgam'F3 7$$!3t*****\\#=/8DF3$\"3G+++v\"ep[(F37$$!3=mmm;a*el\"F3$\"3#QLLLe/TM)F 37$$!3komm;Wn(o)F`o$\"39LLLeDBJ\"*F37$$!3$G++]7bDW%F`o$\"3s****\\([Wdb *F37$$!3IqLLL$eV(>!#?$\"3Immm;kD!)**F37$$\"3V[mmT+07UF`o$\"3I]`bOTx,5F 07$$\"3)Qjmm\"f`@')F`o$\"3APi:)3Lu+\"F07$$\"3%z****\\nZ)H;F3$\"3/HqVMS cE5F07$$\"3ckmm;$y*eCF3$\"3?o$=Oul/1\"F07$$\"3f)******R^bJ$F3$\"3]>/'3 \")G*46F07$$\"3'e*****\\5a`TF3$\"3s].aK!>D<\"F07$$\"3'o****\\7RV'\\F3$ \"3Od+[HmWY7F07$$\"3Y'*****\\@fkeF3$\"3p:%e3TMRM\"F07$$\"3_ILLL&4Nn'F3 $\"3]3\\\"\\Hd`W\"F07$$\"3A*******\\,s`(F3$\"3SAg^kS4o:F07$$\"3%[mm;zM )>$)F3$\"3)4F1'4l>#p\"F07$$\"3M*******pfa<*F3$\"3\"3Cjqg!*=%=F07$$\"39 HLLeg`!)**F3$\"3+=p6+56'*>F07$$\"3w****\\#G2A3\"F0$\"3EMgH-EF0$\"3w&)3\"\\D2*fYF0 7$$\"\"#F*$\"\"&F*-%'COLOURG6&%$RGBG$\"#5F,$F*F*F\\\\l-%+AXESLABELSG6$ Q\"x6\"Q\"yFa\\l-%*THICKNESSG6#Fc[l-%%VIEWG6$;F(Fb[l;$!\")F,$\"\"%F*" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 4 " }}{PARA 0 " " 0 "" {TEXT -1 33 "Let f be the function defined by " }{XPPEDIT 18 0 "f(x) = PIECEWISE([1-x, x < 1],[sqrt(x-1), 1 <= x]);" "6#/-%\"fG6#%\"x G-%*PIECEWISEG6$7$,&\"\"\"F-F'!\"\"2F'F-7$-%%sqrtG6#,&F'F-F-F.1F-F'" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 28 "f is continuous everywhe re. " }}{PARA 0 "" 0 "" {TEXT -1 33 "To check that f is continuous at \+ " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 11 " note that " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(f(x),x = 1^`- `) = Limit(``(1-x),x = 1^`-`);" "6#/-%&LimitG6$-%\"fG6#%\"xG/F*)\"\"\" %\"-G-F%6$-%!G6#,&F-F-F*!\"\"/F*)F-F." }{XPPEDIT 18 0 "`` = 0;" "6#/%! G\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(f(x),x = 1^`+`) = Lim it(sqrt(x-1),x = 1^`+`);" "6#/-%&LimitG6$-%\"fG6#%\"xG/F*)\"\"\"%\"+G- F%6$-%%sqrtG6#,&F*F-F-!\"\"/F*)F-F." }{XPPEDIT 18 0 "`` = 0;" "6#/%!G \"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 7 "so that" }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Limit(f(x),x = 1) = 1;" "6 #/-%&LimitG6$-%\"fG6#%\"xG/F*\"\"\"F," }{XPPEDIT 18 0 "`` = f(1);" "6# /%!G-%\"fG6#\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "f := x -> piecewise(x<1,1-x ,x>=1,sqrt(x-1)):'f(x)'=f(x);\nplot(f(x),x=-2..4,y,thickness=2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$,&\"\" \"F-F'!\"\"2F'F-7$*$,&F-F.F'F-#F-\"\"#1F-F'" }}{PARA 13 "" 1 "" {GLPLOT2D 441 271 271 {PLOTDATA 2 "6&-%'CURVESG6$7V7$$!\"#\"\"!$\"\"$F *7$$!3!******\\2<#p=!#<$\"3!******\\2<#pGF07$$!31++D^NUbiUCFY$\"3\"******\\>iUC\"F07$$!3B++]7YY08FY $\"3-++DhkaI6F07$$\"3%z-+++XDn%!#?$\"3s******\\XF`**FY7$$\"3C++++y?#> \"FY$\"3u*******>#z2))FY7$$\"3h****\\(3wY_#FY$\"3S++]7RKvuFY7$$\"3F)** ****HOTq$FY$\"3s,+++P'eH'FY7$$\"3I,+](3\">)*\\FY$\"3q)***\\7*3=+&FY7$$ \"3_,+]isVIiFY$\"3[)***\\PFcpPFY7$$\"3&=++](o:;vFY$\"3;)****\\7VQ[#FY7 $$\"3#>++v$)[op)FY$\"32)***\\i6:.8FY7$$\"3p++DJnhL$*FY$\"39$***\\(oKQm '!#>7$$\"3W*****\\i%Qq**FY$\"3Wb+++v`hHFgp7$$\"3%***\\i]2=j5F0$\"3+&[E $R\"yN^#FY7$$\"3&****\\(QIKH6F0$\"3%))[P'e7:'f$FY7$$\"3#******\\4+p=\" F0$\"3#o>oHO%>BVFY7$$\"3#****\\7:xWC\"F0$\"3X!>Cv2jW%\\FY7$$\"37++]Zn% )o8F0$\"3[(>5[XvK2'FY7$$\"3y******4FL(\\\"F0$\"3q3jv-?=_qFY7$$\"3#)*** *\\d6.B;F0$\"3#[f$=vAB$*yFY7$$\"3(****\\(o3lW#4u7F07$$\"3u*****\\KCnu#F0$\"3wOi%o )pj@8F07$$\"3s***\\(=n#f(GF0$\"3q3N!=uW'p8F07$$\"3P+++!)RO+IF0$\"3>G=) GCUVT\"F07$$\"30++]_!>w7$F0$\"3'H'>oCgje9F07$$\"3O++v)Q?QD$F0$\"3oVe!G #HF,:F07$$\"3G+++5jypLF0$\"3x@S)p-6%R:F07$$\"3<++]Ujp-NF0$\"3mztme7*>e \"F07$$\"3++++gEd@OF0$\"3-l?/Hr7>;F07$$\"39++v3'>$[PF0$\"3)HZI2m0yl\"F 07$$\"37++D6EjpQF0$\"37N$y(***)*Rp\"F07$$\"\"%F*$\"3?x)ov!30K " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 5 " }}{PARA 0 "" 0 "" {TEXT -1 33 "Let g be the function defined by " }{XPPEDIT 18 0 "g(x) = PIECEWIS E([2-x, x < 1],[sqrt(x-1), 1 <= x]);" "6#/-%\"gG6#%\"xG-%*PIECEWISEG6$ 7$,&\"\"#\"\"\"F'!\"\"2F'F.7$-%%sqrtG6#,&F'F.F.F/1F.F'" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 37 "g is continuous on the open intervals " }{XPPEDIT 18 0 "``(-infinity,1)" "6#-%!G6$,$%)infinityG!\"\"\"\"\"" }{TEXT -1 4 " and" }{XPPEDIT 18 0 "``(1,infinity)" "6#-%!G6$\"\"\"%)in finityG" }{TEXT -1 24 " but g discontinuous at " }{XPPEDIT 18 0 "x = 1 ;" "6#/%\"xG\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 40 "We can chek that g is not continuous at " }{XPPEDIT 18 0 "x=1" "6#/%\"xG \"\"\"" }{TEXT -1 40 " with reference to the three conditions." }} {PARA 0 "" 0 "" {TEXT -1 88 "Although the first of the three condition s is satisfied, namely g is defined at 1, with " }{XPPEDIT 18 0 "g(1)= 0" "6#/-%\"gG6#\"\"\"\"\"!" }{TEXT -1 50 ", the second condition is no t satisfied, that is, " }{XPPEDIT 18 0 "Limit(g(x),x=1)" "6#-%&LimitG6 $-%\"gG6#%\"xG/F)\"\"\"" }{TEXT -1 62 " does not exist. This is becaus e the left and right limits of " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"x G" }{TEXT -1 4 " as " }{TEXT 301 1 "x" }{TEXT -1 36 " approaches 1 hav e different values." }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(g(x),x = 1^`-`) = Limit(``(2-x),x = 1^`-`);" "6#/-%&LimitG6$- %\"gG6#%\"xG/F*)\"\"\"%\"-G-F%6$-%!G6#,&\"\"#F-F*!\"\"/F*)F-F." } {XPPEDIT 18 0 "`` = 1;" "6#/%!G\"\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "while" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(g(x),x = 1^`+`) = Limit(sqrt(x-1),x = 1^`+`);" "6#/-%&LimitG6 $-%\"gG6#%\"xG/F*)\"\"\"%\"+G-F%6$-%%sqrtG6#,&F*F-F-!\"\"/F*)F-F." } {XPPEDIT 18 0 "`` = 0;" "6#/%!G\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 345 "g := x - > piecewise(x<1,2-x,x>=1,sqrt(x-1)):'g(x)'=g(x);\np1 := plot(2-x,x=-2. .0.97,y,thickness=2):\np2 := plot(sqrt(x-1),x=1..4,y,thickness=2,disco nt=true):\np3 := plot([[[1,1]]$2],style=point,color=red,symbol=[circle $2],symbolsize=[15,18]):\np4 := plot([[[1,0]]$3],style=point,color=red ,symbol=[circle,diamond,cross]):\nplots[display]([p||(1..4)]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG-%*PIECEWISEG6$7$,&\"\"# \"\"\"F'!\"\"2F'F.7$*$,&F.F/F'F.#F.F-1F.F'" }}{PARA 13 "" 1 "" {GLPLOT2D 441 271 271 {PLOTDATA 2 "6+-%'CURVESG6%7S7$$!\"#\"\"!$\"\"%F *7$$!3#***\\7_CEN>!#<$\"3#***\\7_CENRF07$$!3-](oylM*y=F0$\"3!)\\(oylM* yQF07$$!32+v3i\")e:=F0$\"3&)*\\(3i\")e:QF07$$!31+Dhk<#=v\"F0$\"31+Dhk< #=v$F07$$!3;]PpP%e$)o\"F0$\"3;]PpP%e$)o$F07$$!3+](o(4)>&H;F0$\"3+](o(4 )>&HOF07$$!33](='=efo:F0$\"3')\\(='=efoNF07$$!31](=)R\")e0:F0$\"3%)\\( =)R\")e0NF07$$!3C]PWFDyU9F0$\"3C]PWFDyUMF07$$!3?+]KN#z\"y8F0$\"3?+]KN# z\"yLF07$$!3;+DY?mF@8F0$\"3;+DY?mF@LF07$$!3)***\\Z`x@d7F0$\"3)***\\Z`x @dKF07$$!37+]iae*G>\"F0$\"37+]iae*G>$F07$$!3=+]_'y448\"F0$\"3=+]_'y448 $F07$$!3=](=$)\\?Y2\"F0$\"3=](=$)\\?Y2$F07$$!39+]A!4(o25F0$\"39+]A!4(o 2IF07$$!3'>++!Rr&)4&*!#=$\"3?++!Rr&)4&HF07$$!3G-voO`G]))F[q$\"3A](oO`G ])GF07$$!3[,+]JDXm#)F[q$\"3:++:`_kEGF07$$!3'**\\(o6a*ei(F[q$\"3+](o6a* eiFF07$$!3F,D1bN$f,(F[q$\"38]i]bLf,FF07$$!3=,](oM-&zjF[q$\"37+voM-&zj# F07$$!3w+vVD)f]z&F[q$\"33]Pa#)f]zDF07$$!3+-]i5'fY;&F[q$\"3)**\\i5'fY;D F07$$!3P-v=e4&)4XF[q$\"3Y](=e4&)4X#F07$$!3y-DJ,\"Q)RRF[q$\"3]]785Q)RR# F07$$!3R+]()*f3UK$F[q$\"3\")*\\()*f3UKBF07$$!3w-+]&3.#)o#F[q$\"3]++b3. #)oAF07$$!3/-]Pqd*f1#F[q$\"3?+v.x&*f1AF07$$!3U.vo*>yRY\"F[q$\"37](o*>y RY@F07$$!3#z,]iG.a&z!#>$\"3=+D'G.a&z?F07$$!3u`++:<>\\>Fbu$\"3w++:<>\\> ?F07$$\"3a)***\\P6sjWFbu$\"3-+]i)yi`&>F07$$\"3@%\\7GLxu-\"F[q$\"3c](=n E_s*=F07$$\"3q(**\\s#)yFm\"F[q$\"3B+]F<@sL=F07$$\"3;%\\7L-[0E#F[q$\"3e ](ow>XRx\"F07$$\"3W'\\iD2*Q&)GF[q$\"3O]Pu#4h9r\"F07$$\"3/&**\\(3aG'\\$ F[q$\"3]+]7f9P];F07$$\"3c$\\7yDPe8%F[q$\"3k](=UF;ke\"F07$$\"3)z****4q, =v%F[q$\"3?++!*H)>[_\"F07$$\"3W'*\\()4Vr\"Q&F[q$\"3O+D,p&G=Y\"F07$$\"3 Y(\\7V#4T1gF[q$\"3C](ov!*e$*R\"F07$$\"3)o***\\MAW!e'F[q$\"3J++bwd&>M\" F07$$\"3K'*\\P&*oMQsF[q$\"3Q+DY5`;w7F07$$\"3U(****pm%yEyF[q$\"3E++IL:K <7F07$$\"3m)\\7L1#=a%)F[q$\"39](oOz\"ea6F07$$\"3M'\\(oD9oa!*F[q$\"3O]7 Vd=`%4\"F07$$\"3u*************p*F[q$\"3.++++++I5F0-%'COLOURG6&%$RGBG$ \"#5!\"\"$F*F*Fa[l-%*THICKNESSG6#\"\"#-F$6%7W7$$\"\"\"F*Fa[l7$$\"3,+]i lyM;5F0$\"3OjM'=s(ey7F[q7$$\"3.++DJdpK5F0$\"3eGI%p4'>3=F[q7$$\"3/+](of V!\\5F0$\"3.j7?'*)yX@#F[q7$$\"31++]i9Rl5F0$\"3uEpsVafS*\\\"F0$\"39Ki$*yd'o1( F[q7$$\"35+](=$f%Gc\"F0$\"3]#eM!eeI-vF[q7$$\"3$*****\\#y,\"G;F0$\"3A_p 3msGDzF[q7$$\"37++Dr\"zbo\"F0$\"3m*HI3%3(*z#)F[q7$$\"3%*****\\(4&G]%G(Hn:H!*F[q7$$\"3%*****\\-* oy(=F0$\"3#\\xnl%eYp$*F[q7$$\"35+]PpnsM>F0$\"3Svvudo7o'*F[q7$$\"3,++]s iL-?F0$\"3]uYxau;,5F07$$\"3-+++!R5'f?F0$\"3$yhm`yt$H5F07$$\"3)***\\P/Q BE@F0$\"3#eCfLjT71\"F07$$\"3!******\\\"o?&=#F0$\"3g'4&)>/s')3\"F07$$\" 3%)**\\Pa&4*\\AF0$\"3mO]]RN*z6\"F07$$\"3&)**\\7j=_6BF0$\"3!**[V-&p@X6F 07$$\"33++vVy!eP#F0$\"3=gx5b$[H<\"F07$$\"3K+](=WU[V#F0$\"3I&3v(*3\\y> \"F07$$\"3)****\\7B>&)\\#F0$\"3(GV93?STA\"F07$$\"3)***\\P>:mkDF0$\"3!Q -Q*3V'3D\"F07$$\"3'***\\iv&QAi#F0$\"3CY$3\")HrOF\"F07$$\"31++vtLU%o#F0 $\"3w]oVrL&yH\"F07$$\"37+++bjm[FF0$\"3i>-.*[rBK\"F07$$\"3\"****\\(yb^6 GF0$\"3AOODZb#fM\"F07$$\"3)***\\PMaKsGF0$\"3]mD'>VH$o8F07$$\"3=++D6W%) RHF0$\"3=\"G#\\tHy#R\"F07$$\"3z*****\\@80+$F0$\"39awj**\\R99F07$$\"31+ +]7,HlIF0$\"3;4P#es6rV\"F07$$\"3()**\\P4w)R7$F0$\"3Y!3B_o!Rd9F07$$\"3; ++]x%f\")=$F0$\"3e//1tGCz9F07$$\"3!)**\\P/-a[KF0$\"3Osm]AL^*\\\"F07$$ \"3/+](=Yb;J$F0$\"3TV@zCHT?:F07$$\"3')****\\i@OtLF0$\"3>nTWC?dS:F07$$ \"3')**\\PfL'zV$F0$\"3!H`5j)yRh:F07$$\"3>+++!*>=+NF0$\"3]2!G$zj>\"e\"F 07$$\"3-++DE&4Qc$F0$\"3P#H'RM+>,;F07$$\"3=+]P%>5pi$F0$\"3t,(zhcu2i\"F0 7$$\"39+++bJ*[o$F0$\"3$)Hg!zNk&Q;F07$$\"33++Dr\"[8v$F0$\"3C=lHM)=(e;F0 7$$\"3++++Ijy5QF0$\"33'\\$f()*Rln\"F07$$\"31+]P/)fT(QF0$\"3i'=/vrM`p\" F07$$\"31+]i0j\"[$RF0$\"3HPp3d088 " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Examp le 6 " }}{PARA 0 "" 0 "" {TEXT -1 33 "Let h be the function defined by " }{XPPEDIT 18 0 "h(x) = PIECEWISE([1-x, x < 1],[2, x = 1],[sqrt(x-1) , 1 < x]);" "6#/-%\"hG6#%\"xG-%*PIECEWISEG6%7$,&\"\"\"F-F'!\"\"2F'F-7$ \"\"#/F'F-7$-%%sqrtG6#,&F'F-F-F.2F-F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 37 "h is continuous on the open intervals" }{XPPEDIT 18 0 "``(-infinity,1)" "6#-%!G6$,$%)infinityG!\"\"\"\"\"" }{TEXT -1 4 " and " }{XPPEDIT 18 0 "``(1,infinity)" "6#-%!G6$\"\"\"%)infinityG" }{TEXT -1 24 " but g discontinuous at " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\" \"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 40 "We can chek that h is not continuous at " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 40 " with reference to the three conditions." }}{PARA 0 "" 0 "" {TEXT -1 79 "The first of the three conditions is satisfied, namely h \+ is defined at 1, with " }{XPPEDIT 18 0 "h(1) = 2;" "6#/-%\"hG6#\"\"\" \"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 49 "The second cond ition is also satisfied, that is, " }{XPPEDIT 18 0 "Limit(h(x),x = 1); " "6#-%&LimitG6$-%\"hG6#%\"xG/F)\"\"\"" }{TEXT -1 54 " exists. This is because the left and right limits of " }{XPPEDIT 18 0 "h(x);" "6#-%\" hG6#%\"xG" }{TEXT -1 4 " as " }{TEXT 300 1 "x" }{TEXT -1 34 " approach es 1 have the same value:" }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Limit(h(x),x = 1^`-`) = Limit(``(1-x),x = 1^`-`);" "6#/ -%&LimitG6$-%\"hG6#%\"xG/F*)\"\"\"%\"-G-F%6$-%!G6#,&F-F-F*!\"\"/F*)F-F ." }{XPPEDIT 18 0 "`` = 0;" "6#/%!G\"\"!" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 3 "and" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(h(x),x = 1^`+`) = Limit(sqrt(x-1),x = 1^`+`);" "6#/-%&Limi tG6$-%\"hG6#%\"xG/F*)\"\"\"%\"+G-F%6$-%%sqrtG6#,&F*F-F-!\"\"/F*)F-F." }{XPPEDIT 18 0 "`` = 0;" "6#/%!G\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 52 "However, the third condition is not satisfied since \+ " }{XPPEDIT 18 0 "Limit(h(x),x = 1)<>h(1)" "6#0-%&LimitG6$-%\"hG6#%\"x G/F*\"\"\"-F(6#F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 408 "h := x -> piecewise(x<1,1-x ,x=1,2,x>1,sqrt(x-1)):'h(x)'=h(x);\np1 := plot(1-x,x=-2..0.97,y,thickn ess=2):\np2 := plot(sqrt(x-1),x=1.001..4,y,thickness=2,discont=true): \np3 := plot([[[1,0]]$2],style=point,color=red,symbol=[circle$2],symbo lsize=[15,18]):\np4 := plot([[[1,2]]$4],style=point,color=red,symbol=[ circle$2,diamond,cross],\n symbolsize=[15,10$3]):\nplots[ display]([p||(1..4)],tickmarks=[6,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"hG6#%\"xG-%*PIECEWISEG6%7$,&\"\"\"F-F'!\"\"2F'F-7$\"\"#/F'F -7$*$,&F-F.F'F-#F-F12F-F'" }}{PARA 13 "" 1 "" {GLPLOT2D 441 271 271 {PLOTDATA 2 "6--%'CURVESG6%7S7$$!\"#\"\"!$\"\"$F*7$$!3#***\\7_CEN>!#<$ \"3#***\\7_CENHF07$$!3-](oylM*y=F0$\"3!)\\(oylM*yGF07$$!32+v3i\")e:=F0 $\"3&)*\\(3i\")e:GF07$$!31+Dhk<#=v\"F0$\"31+Dhk<#=v#F07$$!3;]PpP%e$)o \"F0$\"3;]PpP%e$)o#F07$$!3+](o(4)>&H;F0$\"3+](o(4)>&HEF07$$!33](='=efo :F0$\"3')\\(='=efoDF07$$!31](=)R\")e0:F0$\"3%)\\(=)R\")e0DF07$$!3C]PWF DyU9F0$\"3C]PWFDyUCF07$$!3?+]KN#z\"y8F0$\"3?+]KN#z\"yBF07$$!3;+DY?mF@8 F0$\"3;+DY?mF@BF07$$!3)***\\Z`x@d7F0$\"3)***\\Z`x@dAF07$$!37+]iae*G>\" F0$\"37+]iae*G>#F07$$!3=+]_'y448\"F0$\"3=+]_'y448#F07$$!3=](=$)\\?Y2\" F0$\"3=](=$)\\?Y2#F07$$!39+]A!4(o25F0$\"39+]A!4(o2?F07$$!3'>++!Rr&)4&* !#=$\"3?++!Rr&)4&>F07$$!3G-voO`G]))F[q$\"3A](oO`G])=F07$$!3[,+]JDXm#)F [q$\"3:++:`_kE=F07$$!3'**\\(o6a*ei(F[q$\"3+](o6a*eiyRY\"F[q$\"3M](o*>yRY6F07$$!3#z,]iG.a&z!#>$ \"3=+D'G.a&z5F07$$!3u`++:<>\\>Fbu$\"3a++:<>\\>5F07$$\"3a)***\\P6sjWFbu $\"39++D')yi`&*F[q7$$\"3@%\\7GLxu-\"F[q$\"3!e](=nE_s*)F[q7$$\"3q(**\\s #)yFm\"F[q$\"3I-+vs6AP$)F[q7$$\"3;%\\7L-[0E#F[q$\"3%e](ow>XRxF[q7$$\"3 W'\\iD2*Q&)GF[q$\"3c.vVF4h9rF[q7$$\"3/&**\\(3aG'\\$F[q$\"3&\\+]7f9P]'F [q7$$\"3c$\\7yDPe8%F[q$\"3W1v=UF;keF[q7$$\"3)z****4q,=v%F[q$\"3+-++*H) >[_F[q7$$\"3W'*\\()4Vr\"Q&F[q$\"3c.]7!p&G=YF[q7$$\"3Y(\\7V#4T1gF[q$\"3 a-vov!*e$*RF[q7$$\"3)o***\\MAW!e'F[q$\"37.+]lxb>MF[q7$$\"3K'*\\P&*oMQs F[q$\"3o.]i/JlhFF[q7$$\"3U(****pm%yEyF[q$\"3e-++L`@t@F[q7$$\"3m)\\7L1# =a%)F[q$\"3M,voOz\"ea\"F[q7$$\"3M'\\(oD9oa!*F[q$\"3eO]7Vd=`%*Fbu7$$\"3 u*************p*F[q$\"3o-++++++IFbu-%'COLOURG6&%$RGBG$\"#5!\"\"$F*F*Fa [l-%*THICKNESSG6#\"\"#-F$6%7U7$$\"3*)***********4+\"F0$\"3)\\?o,mxA;$F bu7$$\"3W;HnK[oL5F0$\"3/M(*4?GMN=F[q7$$\"3ALeMl'pj1\"F0$\"39'Rf;2Jid#F [q7$$\"3O(o*[j&3[4\"F0$\"37&QN^#**4zIF[q7$$\"3]TNjhuCB6F0$\"3+6WC%yh1^ $F[q7$$\"3C$3AA\\7s=\"F0$\"3hu4W\"*e!oK%F[q7$$\"3I$eaX_,;D\"F0$\"3(exj &y'*)f,&F[q7$$\"3bTg0FXo:8F0$\"3#>VHC-\"e=cF[q7$$\"3G3-dtw4v8F0$\"39dg R1=_ChF[q7$$\"3*[(=WXlhO9F0$\"3\\\\]69mp2mF[q7$$\"3P3_t]%R-]\"F0$\"3)Q d%e\"ygF2(F[q7$$\"3#\\PklJeOc\"F0$\"3APx6R\"=x](F[q7$$\"3im\"R_T#*)G;F 0$\"3l%R\\dtt-$zF[q7$$\"3I$ef[k]jo\"F0$\"3=!R\\W?GYG)F[q7$$\"3))*\\nC+ N5v\"F0$\"3w#e9scFim)F[q7$$\"3++DEg\\)f\"=F0$\"3b5\"zde)=L!*F[q7$$\"31 +D`zidy=F0$\"3o())*H#**RKP*F[q7$$\"3@3_y$>:a$>F0$\"3/_Td!['or'*F[q7$$ \"3Mm\"4/;-I+#F0$\"3du;Fb**\\,5F07$$\"3cmm`'=d-1#F0$\"3)3,Zt!zoH5F07$$ \"3(\\(o5$Reo7#F0$\"3S`/?le`h5F07$$\"3%omhgu6e=#F0$\"3]4&Qj$)\\*)3\"F0 7$$\"3Mvo&y\"H\\]AF0$\"39=&*e1WD=6F07$$\"3/D1<*o%37BF0$\"3q2@*Glia9\"F 07$$\"3y;a8T#\\jP#F0$\"3/]JC<\"zJ<\"F07$$\"3qT51hTONCF0$\"3O(G>L%o1)> \"F07$$\"3c$eR[s>!*\\#F0$\"3oZ\"G.hWVA\"F07$$\"3_3_`l*R^c#F0$\"3;WxaSb 0^7F07$$\"3\"\\7L%HypAEF0$\"3![#Rws:&QF\"F07$$\"3=$3#H**='[o#F0$\"3U0' zIIA!)H\"F07$$\"3m**\\@mM3\\FF0$\"3z)y`H>HDK\"F07$$\"37]PAS<\">\"GF0$ \"3_Ox&)4F2Y8F07$$\"3)\\(o*eK,F(GF0$\"3C()[CznYo8F07$$\"3z\\iW'z(>SHF0 $\"3+7[pI)4HR\"F07$$\"3[m;Exj%3+$F0$\"3K\"or-x7XT\"F07$$\"3')*\\iCo,c1 $F0$\"3@hhd\")R1a\"F07$$\"3\"\\ (=#\\q]\"QMF0$\"3c)=\\!yyXh:F07$$\"3fLL+'f[.]$F0$\"3]gq(>1\\7e\"F07$$ \"3G$euI#\\&Rc$F0$\"3j=5\\OaB,;F07$$\"3mTNkdX.FOF0$\"37ip5JH\"3i\"F07$ $\"3++]h!>)*\\o$F0$\"3d#>.(3kfQ;F07$$\"3k;H>b5V^PF0$\"3!H(RV=Que;F07$$ \"3QLLA,%\\3\"QF0$\"33^]U(zeln\"F07$$\"39vo5^ " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "Limit(h(x),x=0)" "6#-%&LimitG6$-%\"hG6#%\"xG/F)\"\" !" }{TEXT -1 30 " exists, the discontinuity at " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 16 " is removable. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 31 "The greatest integer function " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Consider the function f d efined by " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)=`` " "6#/-%\"fG6#%\"xG%!G" }{TEXT -1 20 "the greatest integer" }{XPPEDIT 18 0 "``<=x" "6#1%!G%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 61 "We could use double square brackets to denote this function. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "Thus, f or example, " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(2.3 4567) = [[2.34567]];" "6#/-%\"fG6#-%&FloatG6$\"'nXB!\"&7#7#-F(6$F*F+" }{XPPEDIT 18 0 "``=2" "6#/%!G\"\"#" }{TEXT -1 2 ", " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(-2.34567) = [[-2.34567]];" "6#/-% \"fG6#,$-%&FloatG6$\"'nXB!\"&!\"\"7#7#,$-F)6$F+F,F-" }{XPPEDIT 18 0 "` `=-3" "6#/%!G,$\"\"$!\"\"" }{TEXT -1 2 ", " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(Pi) = [[Pi]];" "6#/-%\"fG6#%#PiG7#7#F'" } {XPPEDIT 18 0 "``=3" "6#/%!G\"\"$" }{TEXT -1 2 ", " }}{PARA 257 "" 0 " " {TEXT -1 2 " " }{XPPEDIT 18 0 "f(-Pi)=[[-Pi]]" "6#/-%\"fG6#,$%#PiG! \"\"7#7#,$F(F)" }{XPPEDIT 18 0 "``=-4" "6#/%!G,$\"\"%!\"\"" }{TEXT -1 2 ", " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(sqrt(101)) =[[sqrt(101)]]" "6#/-%\"fG6#-%%sqrtG6#\"$,\"7#7#-F(6#F*" }{XPPEDIT 18 0 "``=10" "6#/%!G\"#5" }{TEXT -1 2 ", " }}{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+,$F-F.F)" }{XPPEDIT 18 0 "` `=56" "6#/%!G\"#c" }{TEXT -1 2 ", " }}{PARA 257 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "f(56-10^(-100)) = [[56-10^(-100)]];" "6#/-%\"fG6#,& \"#c\"\"\")\"#5,$\"$+\"!\"\"F.7#7#,&F(F))F+,$F-F.F." }{XPPEDIT 18 0 "` `=55" "6#/%!G\"#b" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 14 "The value of " }{XPPEDIT 18 0 "[[x]]" "6 #7#7#%\"xG" }{TEXT -1 35 " \"jumps up\" from one integer value " } {XPPEDIT 18 0 "n-1" "6#,&%\"nG\"\"\"F%!\"\"" }{TEXT -1 21 " to the nex t integer " }{TEXT 272 1 "n" }{TEXT -1 4 " as " }{TEXT 270 1 "x" } {TEXT -1 37 " increases through the integer value " }{TEXT 271 1 "n" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 33 "More precisely, for eac h integer " }{TEXT 273 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 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}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "The function f given by " }{XPPEDIT 18 0 "f(x)=[[x]]" "6# /-%\"fG6#%\"xG7#7#F'" }{TEXT -1 123 " is discontinuous at every intege r, but is continuous everywhere else, that is, at all real numbers tha t are not integers. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 54 "For example, we can check that f is not continuous at \+ " }{XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" }{TEXT -1 48 " with reference t o the 3 conditions as follows. " }}{PARA 15 "" 0 "" {TEXT -1 20 "(1) f is defined at " }{XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" }{TEXT -1 5 " an d " }{XPPEDIT 18 0 "f(2)=2" "6#/-%\"fG6#\"\"#F'" }{TEXT -1 1 "." }} {PARA 15 "" 0 "" {TEXT -1 4 "(2) " }{XPPEDIT 18 0 "Limit(f(x),x = 2^`- `) = 1;" "6#/-%&LimitG6$-%\"fG6#%\"xG/F*)\"\"#%\"-G\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Limit(f(x),x=2^`+`)=2" "6#/-%&LimitG6$-%\"fG 6#%\"xG/F*)\"\"#%\"+GF-" }{TEXT -1 4 " so " }{XPPEDIT 18 0 "Limit(f(x) ,x=2)" "6#-%&LimitG6$-%\"fG6#%\"xG/F)\"\"#" }{TEXT -1 16 " does not ex ist." }}{PARA 15 "" 0 "" {TEXT -1 41 "(3) inapplicable because of cond ition 2. " }}{PARA 0 "" 0 "" {TEXT -1 24 " f is not continuous at " } {XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" }{TEXT -1 34 " because the 2nd con dition fails. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 67 "Conti nuity on general intervals and classes of continuous functions" }} {PARA 0 "" 0 "" {TEXT -1 24 "The function f given by " }{XPPEDIT 18 0 "f(x) = sqrt(4-x^2);" "6#/-%\"fG6#%\"xG-%%sqrtG6#,&\"\"%\"\"\"*$F'\"\" #!\"\"" }{TEXT -1 35 " is defined on the closed interval " }{XPPEDIT 18 0 "[-2, 2];" "6#7$,$\"\"#!\"\"F%" }{TEXT -1 43 " and its values alw ays change gradually as " }{TEXT 269 1 "x" }{TEXT -1 10 " changes. " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "A functi on f is said to be continuous on a closed interval " }{XPPEDIT 18 0 "[ a,b]" "6#7$%\"aG%\"bG" }{TEXT -1 41 " if it is continuous on the open \+ interval" }{XPPEDIT 18 0 "``(a,b)" "6#-%!G6$%\"aG%\"bG" }{TEXT -1 10 " and also " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(f (x),x=a^`+`)=f(a)" "6#/-%&LimitG6$-%\"fG6#%\"xG/F*)%\"aG%\"+G-F(6#F-" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "Limit(f(x),x=b^`-`)=f(b)" "6#/-% &LimitG6$-%\"fG6#%\"xG/F*)%\"bG%\"-G-F(6#F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 53 "These last two conditions express the fact that f is " }{TEXT 261 25 "continuous from the right" }{TEXT -1 4 " at " } {XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG" }{TEXT -1 5 " and " }{TEXT 261 24 "continuous from the left" }{TEXT -1 4 " at " }{XPPEDIT 18 0 "x=b" "6# /%\"xG%\"bG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "The function f given by " }{XPPEDIT 18 0 "f(x)= sqrt(4-x^2)" "6#/-%\"fG6#%\"xG-%%sqrtG6#,&\"\"%\"\"\"*$F'\"\"#!\"\"" } {TEXT -1 33 " is continuous from the right at " }{XPPEDIT 18 0 "x=-2" "6#/%\"xG,$\"\"#!\"\"" }{TEXT -1 33 " and continuous from the left at \+ " }{XPPEDIT 18 0 "x=2" "6#/%\"xG\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 19 "This follows since " }{XPPEDIT 18 0 "Limit(f(x),x = \+ -2^`+`) = 0;" "6#/-%&LimitG6$-%\"fG6#%\"xG/F*,$)\"\"#%\"+G!\"\"\"\"!" }{XPPEDIT 18 0 "``=f(-2)" "6#/%!G-%\"fG6#,$\"\"#!\"\"" }{TEXT -1 5 " a nd " }{XPPEDIT 18 0 "Limit(f(x),x = 2^`-`) = 0;" "6#/-%&LimitG6$-%\"fG 6#%\"xG/F*)\"\"#%\"-G\"\"!" }{XPPEDIT 18 0 "``=f(2" "6#/%!G-%\"fG6#\" \"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 45 "Hence f is contin uous on the closed interval " }{XPPEDIT 18 0 "[-2,2]" "6#7$,$\"\"#!\" \"F%" }{TEXT -1 36 " which constitutes the domain of f. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "g := x -> sqrt(4-x^2): 'g(x)'=g(x);\nplot(g(x),x=-2..2,y,thickness=2,view= [-2.3..2.3,0..2.3],scaling=constrained);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG*$,&\"\"%\"\"\"*$)F'\"\"#F+!\"\"#F+F." }}{PARA 13 "" 1 "" {GLPLOT2D 357 315 315 {PLOTDATA 2 "6'-%'CURVESG6$7_o7$$!\"#\" \"!$F*F*7$$!3SL$e9r]X*>!#<$\"35'GV9$*z`Z\"!#=7$$!3#om;HU,\"*)>F/$\"3-b ^-S'y]3#F27$$!3A+]PM@l$)>F/$\"3'))*\\c[X%>b#F27$$!3SLL$e%G?y>F/$\"3P7 \\Z=crWHF27$$!3)****\\(oUIn>F/$\"3@pQd$z\"e,OF27$$!3ymmm\"p0k&>F/$\"3A XE1hA.`TF27$$!3&*****\\P&3Y$>F/$\"3M\"oFHEuB2&F27$$!3MLLL$Q6G\">F/$\"3 #*o0<(4F3%eF27$$!31++v3-)[(=F/$\"3ur>(=$H'pF27$$!3bmm;M!\\p$=F/$\"3 \\(ziTJ.'4zF27$$!3#)***\\7Y\"H%z\"F/$\"3A#R\\y$QpM))F27$$!3MLLL))Qj^J$>4nU49F/7$$!3SLL$3WDTL\"F/$\"3U^t_BJ+!\\\"F/7$$!35++]d(Q &\\7F/$\"3)yCotn=;c\"F/7$$!3gmmmc4`i6F/$\"3uw9,jzUF;F/7$$!3KLLLQW*e3\" F/$\"35zMAYO`z;F/7$$!3w++++()>'***F2$\"3IHh6?-FK=F/7$$!3\\LLL3k(p`(F2$\" 3ibgOe&\\D&=F/7$$!3Anmmmj^NmF2$\"3-Y&Q'>lr')=F/7$$!3)zmmmYh=(eF2$\"3`[ i$e)4'=\">F/7$$!3+,++v#\\N)\\F2$\"3d7A#Qo:p$>F/7$$!3commmCC(>%F2$\"3s# Q#R!)>Yb>F/7$$!39*****\\FRXL$F2$\"3UD#Rk?1?(>F/7$$!3t*****\\#=/8DF2$\" 3'QZCjt[T)>F/7$$!3=mmm;a*el\"F2$\"3=Op\"yBLJ*>F/7$$!3komm;Wn(o)!#>$\"3 O\"\\**o@7\")*>F/7$$!3IqLLL$eV(>!#?$\"3WbqZD!*****>F/7$$\"3)Qjmm\"f`@' )Ffu$\"3!y9kQ'39)*>F/7$$\"3%z****\\nZ)H;F2$\"3M!>y)GzM$*>F/7$$\"3ckmm; $y*eCF2$\"3?x!Q:+E[)>F/7$$\"3f)******R^bJ$F2$\"3]9BTRjKs>F/7$$\"3'e*** **\\5a`TF2$\"3kX1Z!*[Rc>F/7$$\"3'o****\\7RV'\\F2$\"3w3v=K*3u$>F/7$$\"3 Y'*****\\@fkeF2$\"3!)p%Hd4%37>F/7$$\"3_ILLL&4Nn'F2$\"3;6;f6gP&)=F/7$$ \"3A*******\\,s`(F2$\"3%\\!)*\\+/a_=F/7$$\"3%[mm;zM)>$)F2$\"30mED%yO(= =F/7$$\"3M*******pfa<*F2$\"3?=p6Hq5xo\"F/7$$\"3;LLL$)G[k6F/$\"31R< Wh=.E;F/7$$\"3#)****\\7yh]7F/$\"3CrH1`Yvg:F/7$$\"3xmmm')fdL8F/$\"3%>'o q8\\\\!\\\"F/7$$\"3bmmm,FT=9F/$\"3ww2k<>+59F/7$$\"3FLL$e#pa-:F/$\"3SIG s'4#)*>8F/7$$\"3!*******Rv&)z:F/$\"3!RdD\\A(RE7F/7$$\"3ILLLGUYo;F/$\"3 ECg:wo#G5\"F/7$$\"3_mmm1^rZF/$\"3Iil>BruJeF27$$\"31+]i0j\"[$>F/$\"3=dKY SLWk]F27$$\"3/++v.Uac>F/$\"3C]tQ%)=]YTF27$$\"3/+D\"G:3u'>F/$\"3Mn)[F/$\"38e1=p$f+%HF27$$\"39]iSwSq$)>F/$\"3a%Qe.z.za #F27$$\"3-+v$40O\"*)>F/$\"3%R())p#es<3#F27$$\"3!*\\(oa-oX*>F/$\"3/PAj) RPIZ\"F27$$\"\"#F*F+-%'COLOURG6&%$RGBG$\"#5!\"\"F+F+-%*THICKNESSG6#Fa` l-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6$Q\"x6\"Q\"yFdal-%%VIEWG6$; $!#BFh`l$\"#BFh`l;F+F\\bl" 1 2 0 1 10 2 2 9 1 4 1 1.000000 45.000000 46.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "Cont inuity on an half-open intervals is defined in a similar way." }} {PARA 0 "" 0 "" {TEXT -1 34 "For example, the function g where " } {XPPEDIT 18 0 "g(x)=sqrt(x)" "6#/-%\"gG6#%\"xG-%%sqrtG6#F'" }{TEXT -1 32 " is continuous on the interval [" }{XPPEDIT 18 0 "0,infinity" "6$ \"\"!%)infinityG" }{TEXT -1 39 ") since g is continuous on the interva l" }{XPPEDIT 18 0 " ``(0,infinity)" "6#-%!G6$\"\"!%)infinityG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Limit(g(x),x = 0^`+`) = 0;" "6#/-%&LimitG 6$-%\"gG6#%\"xG/F*)\"\"!%\"+GF-" }{XPPEDIT 18 0 "`` = g(0);" "6#/%!G-% \"gG6#\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 75 "A wide va riety of functions are continuous at every point in their domain. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Any " } {TEXT 261 19 "polynomial function" }{TEXT -1 8 " p where" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "p(x)=a[n]*x^n+a[n-1]*x^(n-1)+a[ n-2]*x^(n-2)+` . . . `+a[2]*x^2+a[1]*x+a[0]" "6#/-%\"pG6#%\"xG,0*&&%\" aG6#%\"nG\"\"\")F'F-F.F.*&&F+6#,&F-F.F.!\"\"F.)F',&F-F.F.F4F.F.*&&F+6# ,&F-F.\"\"#F4F.)F',&F-F.F;F4F.F.%(~.~.~.~GF.*&&F+6#F;F.*$F'F;F.F.*&&F+ 6#F.F.F'F.F.&F+6#\"\"!F." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 36 "is continuous on the whole real line" }{XPPEDIT 18 0 "``(-infinity ,infinity)" "6#-%!G6$,$%)infinityG!\"\"F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Any " }{TEXT 261 17 "rational function" }{TEXT -1 8 " r where" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "r(x)=p (x)/q(x)" "6#/-%\"rG6#%\"xG*&-%\"pG6#F'\"\"\"-%\"qG6#F'!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "p(x)" " 6#-%\"pG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "q(x)" "6#-%\"qG6# %\"xG" }{TEXT -1 78 " are polynomials is continuous wherever it is def ined, that is, at all points " }{TEXT 292 1 "x" }{TEXT -1 11 " such th at " }{XPPEDIT 18 0 "q(x)<>0" "6#0-%\"qG6#%\"xG\"\"!" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 28 "Any radical function s where" }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "s(x)=x^(1/n)" "6#/-% \"sG6#%\"xG)F'*&\"\"\"F*%\"nG!\"\"" }{XPPEDIT 18 0 "``=``" "6#/%!GF$" }{TEXT 275 1 "n" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(x)" "6#-%%sqrtG6 #%\"xG" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 31 "is continuous \+ on the interval [" }{XPPEDIT 18 0 "0,infinity" "6$\"\"!%)infinityG" } {TEXT -1 5 ") if " }{TEXT 276 1 "n" }{TEXT -1 51 " is even, and contin uous on the whole real line if " }{TEXT 277 1 "n" }{TEXT -1 8 " is odd ." }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 261 23 "trigonometric fu nctions" }{TEXT -1 92 ": sin, cos, tan, cot, sec, csc are continuous w herever they are defined. For example, since " }{XPPEDIT 18 0 "csc*x = 1/(sin*x);" "6#/*&%$cscG\"\"\"%\"xGF&*&F&F&*&%$sinGF&F'F&!\"\"" } {TEXT -1 50 " the function csc is defined for all real numbers " } {TEXT 293 1 "x" }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "sin*x<>0" " 6#0*&%$sinG\"\"\"%\"xGF&\"\"!" }{TEXT -1 62 ". These are all real numb ers excluding any number of the form " }{XPPEDIT 18 0 "k*Pi" "6#*&%\"k G\"\"\"%#PiGF%" }{TEXT -1 8 ", where " }{TEXT 294 1 "k" }{TEXT -1 69 " is an integer. Thus the function csc is discontinuous at any number \+ " }{XPPEDIT 18 0 "x=k*Pi" "6#/%\"xG*&%\"kG\"\"\"%#PiGF'" }{TEXT -1 8 " , where " }{TEXT 295 1 "k" }{TEXT -1 79 " is an integer, but is contin uous at all other real numbers (everywhere else). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 332 "pi := evalf (Pi):\np1 := plot(csc(x),x=-1.8..8.3,y=-4..4,color=red,thickness=2,dis cont=true):\np2 := plot([seq([[j*pi,-4],[j*pi,4]],j=1..2)],color=black ,linestyle=3):\nplots[display]([p1,p2],xtickmarks=[-pi/2=`-p/2`,pi/2=` p/2`,pi=`p`,3*pi/2=`3p/2`,\n 2*pi=`2p`,5*pi/2=`5p/2`],font=[SYMBOL,11 ],labels=[`x`,`y`],labelfont=[HELVETICA,10]);" }}{PARA 13 "" 1 "" {GLPLOT2D 578 469 469 {PLOTDATA 2 "6)-%'CURVESG6(7gn7$$!3/+++++++=!#<$ !32Gjp#oao-\"F*7$$!3up%yD7l2w\"F*$!3^qv[O%>$=5F*7$$!3%fu@b1Fms\"F*$!3F 4eo$olA,\"F*7$$!3eHN(*\\_B)o\"F*$!3q!R&H-e$p+\"F*7$$!3;A3b[*)e\\;F*$!3 D%)en)=7J+\"F*7$$!3KZFDEj76;F*$!3hP_%f!Q\"3+\"F*7$$!3#p1C=bmad\"F*$!3# 3-+c!4,+5F*7$$!3A9z*3#HaQ:F*$!3qHqkm._+5F*7$$!35(Gu*[kN+:F*$!3&[**ov-' [-5F*7$$!3/:/bTCHi9F*$!3))zGmhb\"f+\"F*7$$!3:APDJ*QJU\"F*$!3,0v/s7+65F *7$$!33&ps!)\\_')Q\"F*$!36=EsH1#o,\"F*7$$!3CU.SU*G)\\8F*$!3wG9tF+#\\- \"F*7$$!3A3$yM(f%3J\"F*$!3JGm$zFlZ.\"F*7$$!3mUMbf'yKF\"F*$!3_4DEk;&f/ \"F*7$$!3.sm\\RR;R7F*$!3)e(Ro-zid5F*7$$!3q.GqP#)f)>\"F*$!3MKF'eY-N2\"F *7$$!3QK:FnPBk6F*$!3G+xsb.$f8F*7$$!3/\"3 8xxfM*yFcr$!3$=b9BTj'39F*7$$!3S(*R)4*=+3vFcr$!3)e$3$*eTzl9F*7$$!3M(=Q# \\l!48(Fcr$!3$*GZTTTkG:F*7$$!3G1z@;u/mnFcr$!393B!*yI1(f\"F*7$$!3OM\"yd bL4O'Fcr$!3Ov.%y/RLo\"F*7$$!3Uhh+M2#p*fFcr$!3Es$3DxH=x\"F*7$$!31#[$y\\ $f#3cFcr$!3e&puU&G5!)=F*7$$!35`yBpV2c_Fcr$!3wJR*)G*pI*>F*7$$!3/8zDhJ/r [Fcr$!3vXoTCtVO@F*7$$!39%[K2!)e(3XFcr$!31)Qgu_s[H#F*7$$!3hm)*[cs1ITFcr $!3W:S]Ou\\\"\\#F*7$$!3%oM![`q#)fPFcr$!37L0p#p4Ms#F*7$$!3#>c0I()>AP$Fc r$!3O1$)=lSOAIF*7$$!3g$=-+43*)*HFcr$!3qKQCTP&3K&F*7$$!3sxh^05\">\\\"Fcr$!3pHf&3eUxs'F*7$$!38O%HP0#GN6Fcr$!31' f1MhLt#))F*7$$!3ir\"*R#37/b(!#>$!3#*z#*pi-pD8!#;7$$!3Xp&Q&[qrIdF\\z$!3 %Qq\"pF\\z$!3^9)=!>$yS6&F_z7$$!3iPU]![LmY\" F\\z$!3RqeBT0e=oF_z7$$!3%y\"\\pOxbx(*!#?$!3@Ot,&emF-\"!#:7$$!3#y=@D?pJ L(F\\\\l$!3Ecr`1\"zOO\"F_\\l7$$!3oeuMo1y))[F\\\\l$!39V(pE(z]X?F_\\l7$$ !35%fg7S'emOF\\\\l$!3h$G36aQts#F_\\l7$$!3_HPA7F\\\\l$!3CALdgK)>=)F_\\l7$$!3)**************f$!#E$!3 #zxxxxxxx#!\"*7ao7$$\"3y*****zI&=$G'F\\^l$\"3KS6rI%\\:f\"F_^l7$$\"3ow< K\")o$*R@F\\\\l$\"3i+aj#fQIn%F_\\l7$$\"3QX#e%zu')zUF\\\\l$\"3vlpNw!GlL #F_\\l7$$\"3k8Zfx!)z>kF\\\\l$\"3Yr5!34#pd:F_\\l7$$\"3!>=JdnG(f&)F\\\\l $\"3QO7]&fv#o6F_\\l7$$\"3%=T+s)*eRG\"F\\z$\"3%)z\"[SwC')y(F_z7$$\"3]0x #o5X>r\"F\\z$\"3$yhUO7%fTeF_z7$$\"3!GH#3Yt\"zc#F\\z$\"3-z9PbVj%*QF_z7$ $\"3Y!)oL&e*)QU$F\\z$\"3ft89dcA@HF_z7$$\"32bg%Q1Me8&F\\z$\"3gEK9U'fz%> F_z7$$\"3QI_NU&yx%oF\\z$\"3GH#40_p9Y\"F_z7$$\"37E@U;g)o#)*F\\z$\"3'H&z jBgD>5F_z7$$\"3K-*[!\\$*f!G\"Fcr$\"3!\\H$G/yAIyF*7$$\"3a%R\"*\\$Hm]>Fc r$\"33[!)Qd#=\"f^F*7$$\"3+qcLP\"o^i#Fcr$\"3!f*[bppQ`QF*7$$\"3!yi$>MvY' H$Fcr$\"3#[%)yi+'>*3$F*7$$\"3N)=&Q2![)=RFcr$\"38FM8ERF=EF*7$$\"3/:6pPp GjXFcr$\"3_ls^unMpAF*7$$\"3EAHH,mwH_Fcr$\"3ltcf$yd@+#F*7$$\"3y(*4;c)3T *eFcr$\"3g\"Hh=Ax*)z\"F*7$$\"35.fLBlYxlFcr$\"3A!Hov*QwN;F*7$$\"3?#*GaG $o$zrFcr$\"3RtO%4dW,_\"F*7$$\"3MH6:,s'p&yFcr$\"3+;6VG:z89F*7$$\"3$QlK) f$[t`)Fcr$\"3nN!*4!G*Fcr$\"3E#3)H6!zvD\"F*7$$\"3 )zh7^'eV)y*Fcr$\"3&=!**>#=L]?\"F*7$$\"3I0QN#4W'\\5F*$\"3y@>>f_2`6F*7$$ \"3*)z+s29i46F*$\"3'[(e$3$ol;6F*7$$\"3EjA%QG*Qz6F*$\"3;8V;QT\"=3\"F*7$ $\"3;f?#fnX6C\"F*$\"3aE^*Gt3p0\"F*7$$\"3X*>&*eA-*38F*$\"3NN)o#fCIN5F*7 $$\"3qFz\\%[AMP\"F*$\"3rZ#\\MZ*z>5F*7$$\"3ox-t0EuS9F*$\"3Y4PY)*p^35F*7 $$\"329!*3#\\jD]\"F*$\"3&HI3UQKB+\"F*7$$\"3Qk&Q(pcCp:F*$\"3nd)pA?,++\" F*7$$\"3XN=j6(4&Q;F*$\"3d44DRpH-5F*7$$\"3)Q29PE/))p\"F*$\"3#yWE\"H$\\# 35F*7$$\"3YAhyJS#Rw\"F*$\"3zwYhWM%*=5F*7$$\"3LR%yT7*>J=F*$\"3Cu'z\"=** )[.\"F*7$$\"3o5`8xY,(*=F*$\"37e*p\\Evc0\"F*7$$\"3mkPz3Ypg>F*$\"3?&H&zx @9\"3\"F*7$$\"3@Z$pdJ+9.#F*$\"3bzN>/.M;6F*7$$\"3#zE]#QD$\\4#F*$\"3%*e_ +2\"f]:\"F*7$$\"3)f,IkumF;#F*$\"3p*3Ss]Z]?\"F*7$$\"3L\"e'=@YBCAF*$\"3c %>o$f&[%f7F*7$$\"3#>#))3W_V\"H#F*$\"3ch(4)=p(3L\"F*7$$\"3)e#G<$zlYN#F* $\"3FMdT1m079F*7$$\"3O!=?g$*f2U#F*$\"3\"eV,]WF^^\"F*7$$\"3O()GcT!z`[#F *$\"3dA(e=[=!R;F*7$$\"3I(3zkDHIb#F*$\"3#f!f#zrg7!=F*7$$\"3/`@'QX%==EF* $\"3UPu]G/m+?F*7$$\"3G4+w`]\"[o#F*$\"3EO%=whtsE#F*7$$\"3%zk\\'=R*3v#F* $\"3;(z2W\">\"ei#F*7$$\"3e*z(zKNh6GF*$\"3+78\"\\/)>'3$F*7$$\"3cBl\"G10 7)GF*$\"3[ztwyC<%)QF*7$$\"3(>#>(ob[M%HF*$\"39$)Q*=23+3&F*7$$\"3$=V(z.J \")4IF*$\"3EZQ\"*z'Q/h(F*7$$\"3ZzRYtFdTIF*$\"3Q,o\"R=q9+\"F_z7$$\"37F0 8VCLtIF*$\"314S?(e?hY\"F_z7$$\"3M&*yM%\\(R!4$F*$\"3yg&e8>jT&>F_z7$$\"3 ej_cXDY2JF*$\"3BlT3pK`IHF_z7$$\"3ZZR<\"F_\\l7$$\"3B([$z`K>NJF*$\"3y*ylN@dEc\"F _\\l7$$\"3Kec>&QEt8$F*$\"3D,?_3f(RM#F_\\l7$$\"3'*Gyf;&f%RJF*$\"3Ec+34] $zo%F_\\l7$$\"3/+++[EfTJF*$\"3sT@i\\\\(*)y\"F_^l7ao7$$\"3%)*****pm#fTJ F*$!3%)[-9lm+du!#57$$\"3k#z)e(fKP9$F*$!3'*fQk\\I-tYF_\\l7$$\"3!fex\"GD (e9$F*$!3=Y3q&>ClL#F_\\l7$$\"39zjweC,[JF*$!3o[fLo.pd:F_\\l7$$\"3Ss^N*Q _,:$F*$!3u;U*yiu#o6F_\\l7$$\"3WeF`]AVaJF*$!3C8*p\\Z?')y(F_z7$$\"3]W.r6 @reJF*$!3E^DY<!f?1D7#HF_z7$$\"3oM58,5&H>$F*$!3/z$f0Qfz%>F_z7$$\"3Nz8%eWq+@$F *$!3%*Q:,v$p9Y\"F_z7$$\"3Hu>X?:')RKF*$!3i5(*QafD>5F_z7$$\"3mpD1&f_'pKF *$!3t[u)[SF-$yF*7$$\"3)eDkL&*emL$F*$!3i0'GN4=\"f^F*7$$\"3B\"=/LZ4TS$F* $!33h1z$)oQ`QF*7$$\"3kuop7%R7Z$F*$!3CNF#\\&f>*3$F*7$$\"31(GW(fuZLNF*$! 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" }}{PARA 0 "" 0 "" {TEXT -1 29 "If f and g are continuous at " }{TEXT 296 1 "c" }{TEXT -1 33 ", then the function h defined by " }{XPPEDIT 18 0 "h(x)=f(x)*g( x)" "6#/-%\"hG6#%\"xG*&-%\"fG6#F'\"\"\"-%\"gG6#F'F," }{TEXT -1 23 " is also continuous at " }{XPPEDIT 18 0 "x=c" "6#/%\"xG%\"cG" }{TEXT -1 37 " and so is the function k defined by " }{XPPEDIT 18 0 "k(x) = f(x) /g(x);" "6#/-%\"kG6#%\"xG*&-%\"fG6#F'\"\"\"-%\"gG6#F'!\"\"" }{TEXT -1 15 " provided that " }{XPPEDIT 18 0 "g(c)<>0" "6#0-%\"gG6#%\"cG\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 22 "If g is continuous at \+ " }{TEXT 279 1 "c" }{TEXT -1 24 " and f is continuous at " }{XPPEDIT 18 0 "f(c)" "6#-%\"fG6#%\"cG" }{TEXT -1 11 ", then the " }{TEXT 261 18 "composite function" }{TEXT -1 1 " " }{XPPEDIT 18 0 "h(x)" "6#-%\"h G6#%\"xG" }{TEXT -1 12 " defined by " }{XPPEDIT 18 0 "h(x)=f(g(x))" "6 #/-%\"hG6#%\"xG-%\"fG6#-%\"gG6#F'" }{TEXT -1 18 " is continuous at " } {TEXT 278 1 "c" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{PARA 0 "" 0 " " {TEXT -1 26 "The function f defined by " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)=(sin*x+x)/sqrt(x^2+1)" "6#/-%\"fG6#%\"xG *&,&*&%$sinG\"\"\"F'F,F,F'F,F,-%%sqrtG6#,&*$F'\"\"#F,F,F,!\"\"" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 26 "is continuous everywhere . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "f := x -> (sin(x)+x)/sqrt(x^2+1): 'f(x)'=f(x);\nplot( f(x),x=-15..15,thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"f G6#%\"xG*&,&-%$sinGF&\"\"\"F'F,F,,&*$)F'\"\"#F,F,F,F,#!\"\"F0" }} {PARA 13 "" 1 "" {GLPLOT2D 624 272 272 {PLOTDATA 2 "6&-%'CURVESG6$7cr7 $$!#:\"\"!$!3R-&R:mT5/\"!#<7$$!31++voUIn9!#;$!3=\"H'yA)[h0\"F-7$$!3&** ***\\P&3YV\"F1$!3]u[#>2/c1\"F-7$$!31+Dcc,;19F1$!3ufTy.\\@o5F-7$$!3++]i vB*FR7$ $!3&***\\i&exd7\"F1$!3GoWlz)Hi5*FR7$$!3%***\\7$*y+&4\"F1$!3?')fsJ#\\+0 *FR7$$!3#***\\i+#QU1\"F1$!33jVHp@Py!*FR7$$!3\"***\\iShTK5F1$!3MY5)yD!f )>*FR7$$!3!***\\i!3%f+5F1$!3aZK$y#)zWS*FR7$$!3%****\\7oS:P*F-$!3\"fXnY 2,++\"F-7$$!3s+++v@)*=()F-$!3%[zW'pkRn5F-7$$!3@++DJ_fJ%)F-$!3osGs7#4<4 \"F-7$$!3q****\\(G3U9)F-$!3LF0=.0K46F-7$$!3[**\\(=ZVC)zF-$!3y@&pkMBb6 \"F-7$$!3D***\\ily1#yF-$!3?S\\&)*=(o=6F-7$$!3-**\\iSQ\"*ewF-$!3K(y0w5' f=6F-7$$!3o*****\\-\\r\\(F-$!367'3r.1^6\"F-7$$!3s+++]4HsrF-$!3a*fgSdZw 4\"F-7$$!3'3++](GVZoF-$!3ILb\"eN$ym5F-7$$!3I+++D>PMlF-$!3'\\^=ar\"4E5F -7$$!3u*****\\(4J@iF-$!3!=Yw3\"\\8v(*FR7$$!3!****\\iIKFl&F-$!37bqxQ&p, #))FR7$$!3))**\\i!z%o9`F-$!3w#)Q7!4'z.$)FR7$$!3')*****\\FPm(\\F-$!3CIN '>&>S-zFR7$$!3`++D\")>XL[F-$!3iB;'GVM9y(FR7$$!3I++](om-p%F-$!3Y)4Yk6#[ &p(FR7$$!35++v$R\"3ZXF-$!3y&oJt!=+[wFR7$$!3w++++h*QS%F-$!3'o'Hr%*)R>k( FR7$$!3N]7.Koh?VF-$!3]y/w>jUewFR7$$!3%3]iScPtB%F-$!3-)pz()o,,p(FR7$$!3 K^P4'HeS:%F-$!309JO^ODPxFR7$$!3#4+D\"G!z22%F-$!3X6K!\\](3+yFR7$$!3+,v= #\\?U!RF-$!3m-t)G'))=tzFR7$$!34,+Dc>mPPF-$!3;$e'[?+F4#)FR7$$!3(4+DJ!pz UMF-$!3S!)eNRCfv()FR7$$!3'3+++&=$z9$F-$!3gHS+GoZ6&*FR7$$!3!)***\\iX/4] #F-$!3E%>'4e:X]6F-7$$!3s****\\7H%G>#F-$!3#*=5/;(eqC\"F-7$$!3o***\\(o8y %)=F-$!3MH\"Rg#f8H8F-7$$!3w*\\(=C\"F-$!3?dv*p9RCP\"F-7$$!3y#**\\(=d[n%*FR$!3[72F#z9oF\"F-7$ $!3#f)**\\7ev:lFR$!3_tN^1c,a5F-7$$!3#z)\\7yg$Q#\\FR$!3-f+FA]Ve')FR7$$! 3\"**)*\\PM;>L$FR$!3OGUQU2(RE'FR7$$!3!4\\ilZcf`#FR$!3gAE&Q*y-!*[FR7$$! 3!>*\\P4m**RMFR7$$!3&*G\\(=Un.W*!#>$!3#z\\-)41Ky=FR7$$! 3#*Q****\\(o2[\"Fe]l$!3)zN#=h(e6'HFe]l7$$\"3[]]ils,(y'Fe]l$\"3'HWZh7oP N\"FR7$$\"3*R+D\"G.[0:FR$\"3=Y3E6:zrHFR7$$\"3#H](oH*eAL#FR$\"37A:a8lnH%f2& )FR7$$\"3i(***\\P>:mkFR$\"3G'4w%[A#*[5F-7$$\"3?#*****\\Z+X$*FR$\"3!yXu $*4@/F\"F-7$$\"3o)**\\iv&QA7F-$\"3I21!**yT\"p8F-7$$\"3.\\i!Rv;,I\"F-$ \"3['QmV8G,Q\"F-7$$\"3Q*\\i:vZyP\"F-$\"3'o.b`B!z&Q\"F-7$$\"3u\\(=#\\(y bX\"F-$\"3oPQs=4t'Q\"F-7$$\"35+](ou4L`\"F-$\"3=e<4\\m\\$Q\"F-7$$\"3!3] (=UO=ROkO\"F-7$$\"3_,+]PPBW=F-$\"3D0@o;E=PAg*FR7$$\"3P+]il]?>MF-$\"3PazqD2jG))FR7$$\"3')* **\\PMaKs$F-$\"3j%>tMd3EB)FR7$$\"3W*\\Pfy^?*QF-$\"3!G$))4,`J))zFR7$$\" 3/**\\7G#\\31%F-$\"3OD(y_UK'3yFR7$$\"3G\\(=#\\zCXTF-$\"31d3Yfk:VxFR7$$ \"3j)\\7.nY'HUF-$\"3/*GH&>Q!Qp(FR7$$\"3*zC19RXSJ%F-$\"3Oc%e;`p.m(FR7$$ \"3A)***\\7TW)R%F-$\"3/Xs@*QhDk(FR7$$\"3=)\\(=<,GuWF-$\"3Q&[in'3eRwFR7 $$\"39)*\\(=7;,b%F-$\"3S9drUze[wFR7$$\"35)\\il7_fi%F-$\"3q%\\^$Qt?pwFR 7$$\"31)**\\78)y,ZF-$\"3udF/.!=5q(FR7$$\"3)z*\\iS,Y`[F-$\"3_yq9pzJ'z(F R7$$\"3*y*****\\@80]F-$\"3**)4CJJ>.$zFR7$$\"3x)***\\Pm,H`F-$\"3Mz4[VwF-$\"3`SZJywS=6F-7$$\"3^++DJ$y4!yF-$\"3-rLQT P&)=6F-7$$\"3!4++]Zm(ezF-$\"3JAlp%=#=;6F-7$$\"3G,+v=Yb;\")F-$\"36c9nw[ e56F-7$$\"3c**\\(=7)3D%)F-$\"3U=cQ3/>#4\"F-7$$\"3%y****\\i@Ot)F-$\"3!4 1T\"[/-m5F-7$$\"3Q++v$fL'z$*F-$\"3YHO3v**[\"***FR7$$\"35+++!*>=+5F1$\" 3NYo8Myk2%*FR7$$\"3:+]7ed*>.\"F1$\"3a@*fl)ox+#*FR7$$\"3?++DE&4Q1\"F1$ \"3)>)eu3AQz!*FR7$$\"3?+DJg)f`4\"F1$\"3![!**o,`A]!*FR7$$\"3=+]P%>5p7\" F1$\"33VS!z05)4\"*FR7$$\"3E+vou;!f:\"F1$\"3:IP-HF6M#*FR7$$\"3K+++bJ*[= \"F1$\"3F^Mb<_n6%*FR7$$\"3E++Dr\"[8D\"F1$\"3%pu)ei%4h#**FR7$$\"3#)**** **Hjy58F1$\"30c,DI(4j.\"F-7$$\"3/+v=nIZU8F1$\"3E=)yf'GX`5F-7$$\"3E+]P/ )fTP\"F1$\"3/av>DwLk5F-7$$\"3?+++b!)[/9F1$\"3+)='4lJ>o5F-7$$\"3;+]i0j \"[V\"F1$\"3+]w!3'[cl5F-7$$\"33+D\"G:3uY\"F1$\"39tkcM*3h0\"F-7$$\"#:F* $\"3R-&R:mT5/\"F--%'COLOURG6&%$RGBG$\"#5!\"\"$F*F*F`am-%+AXESLABELSG6$ Q\"x6\"Q!Feam-%*THICKNESSG6#\"\"#-%%VIEWG6$;F(Fe`m%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 134 "The following co mputation uses calculus techniques to obtain approximate coordinates o f some maximum and minimum points on the curve. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 202 "f := x -> ( sin(x)+x)/sqrt(x^2+1):\nx1 := fsolve(D(f)(x),x=1.5): map(evalf[5],(``( x1,f(x1))));\nx2 := fsolve(D(f)(x),x=4): map(evalf[5],``(x2,f(x2)));\n x3 := fsolve(D(f)(x),x=8): map(evalf[5],``(x3,f(x3)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%!G6$$\"&IV\"!\"%$\"&pQ\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%!G6$$\"&\\X%!\"%$\"&#Rw!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%!G6$$\"&Ru(!\"%$\"&\">6F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT -1 4 "L et " }{XPPEDIT 18 0 "f(x)=tan*x" "6#/-%\"fG6#%\"xG*&%$tanG\"\"\"F'F*" }{TEXT -1 8 ". Since " }{XPPEDIT 18 0 "tan*x=sin*x/(cos*x)" "6#/*&%$ta nG\"\"\"%\"xGF&*(%$sinGF&F'F&*&%$cosGF&F'F&!\"\"" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "tan*x" "6#*&%$tanG\"\"\"%\"xGF%" }{TEXT -1 29 " is not \+ defined at values of " }{TEXT 297 1 "x" }{TEXT -1 11 " for which " } {XPPEDIT 18 0 "cos*x=0" "6#/*&%$cosG\"\"\"%\"xGF&\"\"!" }{TEXT -1 19 " . These values are " }{XPPEDIT 18 0 "x = Pi/2+k*Pi;" "6#/%\"xG,&*&%#Pi G\"\"\"\"\"#!\"\"F(*&%\"kGF(F'F(F(" }{TEXT -1 8 ", where " }{TEXT 299 1 "k" }{TEXT -1 16 " is an integer. " }}{PARA 0 "" 0 "" {TEXT -1 129 " The tangent function has discontinuities at these real number values w here it is not defined, but is continuous everywhere else. " }}{PARA 0 "" 0 "" {TEXT -1 50 "For example, it is continuous on the open inter val" }{XPPEDIT 18 0 "``(-Pi/2,Pi/2);" "6#-%!G6$,$*&%#PiG\"\"\"\"\"#!\" \"F+*&F(F)F*F+" }{TEXT -1 14 ". The formula " }{XPPEDIT 18 0 "tan(x+Pi )=tan*x" "6#/-%$tanG6#,&%\"xG\"\"\"%#PiGF)*&F%F)F(F)" }{TEXT -1 76 " w hich expresses the fact that the tangent function is periodic with per iod " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 36 " enables the remaind er of the graph " }{XPPEDIT 18 0 "y=tan*x" "6#/%\"yG*&%$tanG\"\"\"%\"x GF'" }{TEXT -1 68 " to be obtained by translating the continuous branc h on the interval" }{XPPEDIT 18 0 "``(-Pi/2,Pi/2);" "6#-%!G6$,$*&%#PiG \"\"\"\"\"#!\"\"F+*&F(F)F*F+" }{TEXT -1 25 " to intervals of the form " }{XPPEDIT 18 0 "``(-Pi/2+k*Pi,Pi/2+k*Pi);" "6#-%!G6$,&*&%#PiG\"\"\" \"\"#!\"\"F+*&%\"kGF)F(F)F),&*&F(F)F*F+F)*&F-F)F(F)F)" }{TEXT -1 8 ", \+ where " }{TEXT 298 1 "k" }{TEXT -1 66 " is an integer. Hence f is cont inuous on all such open intervals. 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" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "Limit( f(x),x = 0) = Limit(sin*x/x,x = 0);" "6#/-%&LimitG6$-%\"fG6#%\"xG/F*\" \"!-F%6$*(%$sinG\"\"\"F*F1F*!\"\"/F*F," }{XPPEDIT 18 0 "``=1" "6#/%!G \"\"\"" }{XPPEDIT 18 0 "``=f(0)" "6#/%!G-%\"fG6#\"\"!" }{TEXT -1 29 ", f is also continuous at 0. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 237 "pi := evalf(Pi):\nplot(sin( x)/x,x=-14.5..14.5,y=-.25..1.15,color=red,thickness=2,xtickmarks=[-4*p i=`-4p`,\n -3*pi=`-3p`,-2*pi=`-2p`,-pi=`-p`,pi=`p`,2*pi=`2p`,3*pi=`3p `,4*pi=`4p`],\n ytickmarks=3,font=[SYMBOL,11],labelfont=[HELVETICA,10 ]);" }}{PARA 13 "" 1 "" {GLPLOT2D 775 217 217 {PLOTDATA 2 "6)-%'CURVES G6#7is7$$!3+++++++]9!#;$\"3!*zn12@bZk!#>7$$!3hm\"HK1(>M9F*$\"3E1()*f%G #o#oF-7$$!3QL$ek7%R=9F*$\"3uo4zJc^UqF-7$$!3;+vo*=\"f-9F*$\"3;eOXbBe&3( F-7$$!3ymm\"HD)y'Q\"F*$\"3))pd?*3O5&pF-7$$!33]PM^\")Gf8F*$\"3Kgt6%=6J:yF-7$$!3sm;H g0MJ6F*$!3'3O0\\))pjR)F-7$$!3#**\\ibfxp6\"F*$!3H7Y)RWLs\"))F-7$$!3HLL$ 3j9E5\"F*$!3go%*>9Q6l!*F-7$$!3mmT5m;D)3\"F*$!3jLm!R7'QI\"*F-7$$!3%**\\ i!['zL2\"F*$!3'f.m'oE(*)**)F-7$$!3QL3-Iw]e5F*$!3ceyok_Si')F-7$$!3\"o;z >hNO/\"F*$!3K:bD63GA\")F-7$$!33+v$Rfj(G5F*$!3a&*4$GgKYQ(F-7$$!3KL$3FpA +)**!#<$!3%y,Jb'\\1#G&F-7$$!3'emTgW4Cn*Fjq$!3vT(4Z^)4MDF-7$$!3U+](=*f: f!*Fjq$\"3y@S'4M6&HJ$)*p7Fas7$$!3E% ek.N[Xz(Fjq$\"31p^v]Fo!G\"Fas7$$!3]+D\"G-ijr(Fjq$\"313:?qZp$G\"Fas7$$! 3t;/E&pv\"QwFjq$\"3MW_g1[%)y7Fas7$$!3'HL3xO*)*fvFjq$\"3O!>!*y)***fE\"F as7$$!3KmTg7nh.uFjq$\"3wu`\\5:,;7Fas7$$!3o****\\dSCZsFjq$\"3$o*>F4mFjq$\"39$fz8dv6 )\\F-7$$!3d****\\UR$R,'Fjq$!3#3\"fRB`ABWF-7$$!3Mn;/'*yIkaFjq$!3r(e+U#o jO8Fas7$$!3,,v$4jGv8&Fjq$!31$RBPU*=tq+q(4 jo?Fas7$$!3:M3_SJaTZFjq$!3f%o$G.=73@Fas7$$!3]M$3_\"pLsYFjq$!3)pCW5=S&Q @Fas7$$!3&[$e*)*oIJg%Fjq$!3e#RD(H3[f@Fas7$$!3JMLekW#R`%Fjq$!3snnvPUcq@ Fas7$$!3yL3FR#=ZY%Fjq$!3A@fz%RQ9<#Fas7$$!37M$eR,7bR%Fjq$!3m2ZWR/yh@Fas 7$$!3[Mek)y0jK%Fjq$!3**>A\\:#)HT@Fas7$$!3'RLLLc*4dUFjq$!3$\\WY+tJ(4@Fa s7$$!39nT&QR'3NRFjq$!37Nb(QN;9\"=Fas7$$!3M+]PCK28OFjq$!3%[z6wmvV[$ej')GFjq$\"3\"*yuQ'RRpt)F-7$$!3HoTN\")3FIFFjq$\"3Yg&H@B,WY\"F as7$$!3g%eky#f!Rd#Fjq$\"3c%)[8=r(*)3#Fas7$$!3$4+vV(4a@Fjq$\"3%Gg^YD-W -%Fas7$$!3'**\\Pft^3(>Fjq$\"3W>tlhkItYFas7$$!3k**\\7B`&>#=Fjq$\"3O\\k7 dQS;`Fas7$$!3'Ge9m`(fm;Fjq$\"35[)yP(=N0mFas7$$!3M\\Pfj>)eN\"Fjq$\"3&zY\"*eg-c?(Fas7$$!3fKL3xT_+7Fjq$\"3O [l2/A=lxFas7$$!3)>Le9'G!>:*Fas$\"3?p*ej$HNh')Fas7$$!31QL3_Rc)H'Fas$\"3 sHx6LQz^$*Fas7$$!3+sT&)3#3(fZFas$\"3H&fm(ROnE'*Fas7$$!3$f+DcY_3A$Fas$ \"3VLKrCk*z#)*Fas7$$!3!HU5SfC9X#Fas$\"3=41UwC9+**Fas7$$!3')ReRAn*>o\"F as$\"3mX')=rZ\"H&**Fas7$$!3CoD\"y])oD\"*F-$\"3GH\"y(zg7')**Fas7$$!3)yt m;z49V\"F-$\"3!=\")*\\9&e'****Fas7$$\"3k\\2xcLyglF-$\"3%*RA6kv#G***Fas 7$$\"3sB3_]wHb9Fas$\"3938\\s\"RZ'**Fas7$$\"3YsXOlp^aAFas$\"3p,xg@4]:** Fas7$$\"3A@$3-GOP0$Fas$\"3Ed7C;8IX)*Fas7$$\"3s=e*)4\\<_YFas$\"3p([bY:r Jk*Fas7$$\"3A;LeRNh]iFas$\"31[\")ekMVh$*Fas7$$\"3K_mm\"f/N.*Fas$\"3_Er WVvN%p)Fas7$$\"3%))*\\Pk&R;=\"Fjq$\"3uEJa%*o/IyFas7$$\"3!G3xJ\\>>L\"Fj q$\"3mF*R'*ynZH(Fas7$$\"3wm\"z>U*>#[\"Fjq$\"382L&=&)o-s'Fas7$$\"3t]7y] $zCj\"Fjq$\"3@AHTB?+9hFas7$$\"3qMLez#fFy\"Fjq$\"3CS`>!fXP[&Fas7$$\"3[+ v$4:8!Q>Fjq$\"3El&zO[7f\"[Fas7$$\"3Gm;HAqE$4#Fjq$\"3$*Q'yp*p))RTFas7$$ \"33Kek$*3_[AFjq$\"33bK\"QI1XY$Fas7$$\"3(y****\\wuPS#Fjq$\"3?#y&*fj5%) z#Fas7$$\"3E)\\i!*Qgcb#Fjq$\"3)po>o'=tj@Fas7$$\"3l)*\\78ga2FFjq$\"3q*p J'>aB`:Fas7$$\"3/*\\(=P;VfGFjq$\"3kS#oxt9tt*F-7$$\"3V***\\7E<8,$Fjq$\" 3\"\\0+X6mRJ%F-7$$\"3Z)\\PMcJ_I$Fjq$!3Xc))o')*R)G\\F-7$$\"3](*\\ile9*f $Fjq$!3Tl5oplQF7Fas7$$\"3C(\\(=(e([DRFjq$!3[$H_GL7')z\"Fas7$$\"3)p**\\ (3$H=D%Fjq$!3]H$[T,no5#Fas7$$\"3xzXhYu8DVFjq$!3`9zPwz&39#Fas7$$\"3Wj\" zWeX%)R%Fjq$!3qu$Rpc2C;#Fas7$$\"39ZPMAPvrWFjq$!3YJ`Jw_#=<#Fas7$$\"3#*H $3-'=1XXFjq$!3%Q3c4,h%p@Fas7$$\"3r7H2)**p$=YFjq$!3kyn++DG3Al'Fjq$\"3ep(>!**\\KAaF-7$$\"3HKeRi%[S%pFjq$\"3%*))eH fD?R))F-7$$\"3ii;HU'))eB(Fjq$\"3z/DF/&4j7\"Fas7$$\"3!o**\\7=<%)Q(Fjq$ \"3,A#*)*yxT47Fas7$$\"3)4L3-sX4a(Fjq$\"3m$*\\w9&\\;E\"Fas7$$\"32)\\(o* )*4sh(Fjq$\"3'3P9pW*=w7Fas7$$\"3;lm;fUZ$p(Fjq$\"3#f#QvKc4$G\"Fas7$$\"3 EKekG&Q(pxFjq$\"3)HLmUG![#G\"Fas7$$\"3M**\\7)z-g%yFjq$\"3)[7))3x$\\u7F as7$$\"3a:ajCt7&*zFjq$\"3zmf+&eB$Q7Fas7$$\"3sJe9^=DW\")Fjq$\"3'y(3XzU \\w6Fas7$$\"3!zCcwPwLH)Fjq$\"3UaX=#\\V74\"Fas7$$\"35km;/4]U%)Fjq$\"3A? wC'[*4_)*F-7$$\"3w(*\\iS\"zp1*Fjq$\"3%p4@1i8D'QF-7$$\"3iKLL.fUo'*Fjq$! 3O#RW`&y<&\\#F-7$$\"33K$3_**ef(**Fjq$!3'*Rn09xb\\_F-7$$\"3:L$3(3#\\$G5 F*$!3E)[RQjT8O(F-7$$\"3_T&Qo`)fV5F*$!3akx\"e0%o?\")F-7$$\"3))\\(o\\'y% )e5F*$!334GCyYRs')F-7$$\"3Ce*)4$>(4u5F*$!3RNgts\"=+,*F-7$$\"3gm\"H7_Y$ *3\"F*$!33IX:09+K\"*F-7$$\"3*\\(o/q!fL5\"F*$!3)*=2qxToc!*F-7$$\"3@$ek) =;P<6F*$!3dxQ;fQ%z!))F-7$$\"3U\"H#onTQJ6F*$!3-`:&H&4%[R)F-7$$\"3#)**** \\;nRX6F*$!3,%G%e![%HHyF-7$$\"3=L3-Tm^x6F*$!3:D>$H&H-WPF-7$$\"3ILLL_M4n7F*$\"3Jx;$fLNsB)!#?7$$\"3k;z%\\'Rs( H\"F*$\"3'yq36zVx2$F-7$$\"3)**\\ivZa$G8F*$\"3qG;N#*y!z%\\F-7$$\"3#**** *\\'yrwN\"F*$\"314HP5EuQiF-7$$\"3%)*\\Pa4*)pQ\"F*$\"37/Y 0],[0, x = 0]);" " 6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$-%$sinG6#*&\"\"\"F0F'!\"\"0F'\"\"!7$F 3/F'F3" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 63 "general conside rations lead to the fact that f is continuous on" }{XPPEDIT 18 0 " ``( -infinity,0)" "6#-%!G6$,$%)infinityG!\"\"\"\"!" }{TEXT -1 12 " and als o on" }{XPPEDIT 18 0 "``(0,infinity)" "6#-%!G6$\"\"!%)infinityG" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 " Limit(f(x),x = 0) = Limit(sin(1/x),x = 0);" "6#/-%&LimitG6$-%\"fG6#%\" xG/F*\"\"!-F%6$-%$sinG6#*&\"\"\"F3F*!\"\"/F*F," }{TEXT -1 43 " does no t exist, f is not continuous at 0. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 4" }}{PARA 0 "" 0 "" {TEXT -1 21 "Let f be defined by " }}{PARA 257 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "f(x) = PIECEWISE([x*sin(1/x), x <> 0],[0, x = 0]);" "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$*&F'\"\"\"-%$sinG6#*&F-F-F'!\"\"F-0F '\"\"!7$F4/F'F4" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 67 "gener al considerations lead to the fact that f is continuous for on" } {XPPEDIT 18 0 " ``(-infinity,0)" "6#-%!G6$,$%)infinityG!\"\"\"\"!" } {TEXT -1 8 " and for" }{XPPEDIT 18 0 "``(0,infinity)" "6#-%!G6$\"\"!%) infinityG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Now " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-abs(x)<=x*sin(1/x)" "6#1,$-%$absG6#%\"xG!\"\"*&F(\"\"\"-%$sinG6#*&F+F+F(F)F+" }{XPPEDIT 18 0 "``<=abs(x)" "6#1%!G-%$absG6#%\"xG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "for all " }{XPPEDIT 18 0 "x<>0" "6#0%\"xG\"\"!" } {TEXT -1 37 ", so the Squeeze Theorem shows that " }{XPPEDIT 18 0 "Li mit(x*sin(1/x),x = 0) = 0;" "6#/-%&LimitG6$*&%\"xG\"\"\"-%$sinG6#*&F)F )F(!\"\"F)/F(\"\"!F0" }{XPPEDIT 18 0 "`` = f(0);" "6#/%!G-%\"fG6#\"\"! " }{TEXT -1 3 ",. " }}{PARA 0 "" 0 "" {TEXT -1 94 "Hence f is also con tinuous at 0. Hence f is everywhere continuous, that is, f is continuo us on" }{XPPEDIT 18 0 "``(-infinity,infinity)" "6#-%!G6$,$%)infinityG! \"\"F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 "The Intermediate Value Theorem " }}{PARA 0 "" 0 "" {TEXT -1 80 "The following result is an important consequence of properties \+ of real numbers. " }}{PARA 0 "" 0 "" {TEXT -1 60 "If f is a continuou s function defined on a closed interval " }{XPPEDIT 18 0 "[a,b]" "6#7$ %\"aG%\"bG" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "k" "6#%\"kG" }{TEXT -1 26 " is a real number between " }{XPPEDIT 18 0 "f(a)" "6#-%\"fG6#% \"aG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "f(b)" "6#-%\"fG6#%\"bG" } {TEXT -1 48 ", then there is always at least one real number " }{TEXT 287 1 "c" }{TEXT -1 4 " in " }{XPPEDIT 18 0 "[a,b]" "6#7$%\"aG%\"bG" } {TEXT -1 11 " such that " }{XPPEDIT 18 0 "f(c)=k" "6#/-%\"fG6#%\"cG%\" kG" }{TEXT -1 2 ". 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 281 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 48 "First note that f is continuous on the interval " }{XPPEDIT 18 0 "[0,1]" "6#7$\"\"!\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "Now " }{XPPEDIT 18 0 "f(0)=-1" "6#/-%\"fG6#\"\"!,$\"\"\"!\"\"" } {TEXT -1 7 " while " }{XPPEDIT 18 0 "f(1)=2" "6#/-%\"fG6#\"\"\"\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 103 "The number 0 lies bet ween -1 and 2 so the Intermediate Value Theorem guarantees that there \+ is a number " }{TEXT 282 1 "c" }{TEXT -1 37 " somewhere between 0 and \+ 1 such that " }{XPPEDIT 18 0 "f(c)=0" "6#/-%\"fG6#%\"cG\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 11 "The number " }{TEXT 286 1 "c " }{TEXT -1 19 " is then a zero of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6# %\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 43 "From an intuiti ve point of view, the curve " }{XPPEDIT 18 0 "y=x^3+2*x-1" "6#/%\"yG,( *$%\"xG\"\"$\"\"\"*&\"\"#F)F'F)F)F)!\"\"" }{TEXT -1 25 " passes throug h the point" }{XPPEDIT 18 0 "``(0,-1)" "6#-%!G6$\"\"!,$\"\"\"!\"\"" } {TEXT -1 23 ", which lies below the " }{TEXT 283 1 "x" }{TEXT -1 10 " \+ axis, and" }{XPPEDIT 18 0 "``(1,2)" "6#-%!G6$\"\"\"\"\"#" }{TEXT -1 23 ", which lies above the " }{TEXT 284 1 "x" }{TEXT -1 7 " axis. " }} {PARA 0 "" 0 "" {TEXT -1 88 "In drawing the (unbroken) curve from the \+ first point to the second one has to cross the " }{TEXT 285 1 "x" } {TEXT -1 32 " axis somewhere in the interval " }{XPPEDIT 18 0 "[0,1]" "6#7$\"\"!\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot(x^3+2*x-1,x=-.2..1.2,th ickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 486 486 486 {PLOTDATA 2 "6&- %'CURVESG6$7S7$$!35+++++++?!#=$!3#************zS\"!#<7$$!3%omm;%)R[p\" F*$!3=w?Ulj$QM\"F-7$$!3fLLe>;KH9F*$!3Cq-BrVy)G\"F-7$$!3$omm\"4'=28\"F* $!3RK/Ot$*eF7F-7$$!3/ommTEO,$)!#>$!3=Bn\\@$*fm6F-7$$!3aNL$eMD)4`F@$!3] `WV8iM16F-7$$!3))pm;HtGODF@$!3=9V*=1U20\"F-7$$\"3'e&***\\P1bN$!#?$!38V $*o%\\*)G$**F*7$$\"3&)GL$3d4cI$F@$!3>;&p+)o^Q$*F*7$$\"3s(***\\([VhE'F@ $!3KHP>H4JW()F*7$$\"3:lmm;lT6$*F@$!3?5?&yYV'H\")F*7$$\"39LL$eYp$*>\"F* $!3K+#*4$H3Se(F*7$$\"3^*****\\XI8]\"F*$!3Ky*3Y?*\\jpF*7$$\"3Y*****\\KJ X!=F*$!3J]E]&zv@L'F*7$$\"3%)*****\\a@n4#F*$!3mW!3G(*zVr&F*7$$\"3iKL3d# e?O#F*$!3h0zu:n4W^F*7$$\"3-mmmr#pvn#F*$!3OJrm:k*GX%F*7$$\"3Kmmm'[[[%HF *$!3Y[jvg=#\\&QF*7$$\"3w)**\\PvddD$F*$!3At\"y+_vL9$F*7$$\"3clmmO^'4`$F *$!3wE$oQ1Ry\\#F*7$$\"3I***\\PD6H$QF*$!3S%)fs,l2ruAcf5F*7$$\"3&fmmTgO/U%F*$!3%H]*zh*>O&HF@7$$\"3[mmT&RJfp%F*$ \"33&f?&*f))RF%F@7$$\"3'GLLeu*3$*\\F*$\"3B0Q[gR+J7F*7$$\"3uJL3dPv,`F*$ \"3I)GXV*ev$4#F*7$$\"31***\\ioY/d&F*$\"3#zCHx3'RpGF*7$$\"3;KL$3TU1'eF* $\"3O\"3_\"esCMPF*7$$\"31*******)HWghF*$\"3akj!>ER)eYF*7$$\"3m(***\\n$ RPX'F*$\"3;O1I-(4bf&F*7$$\"3z***\\Pp=vt'F*$\"3&>VM'ysZLlF*7$$\"3K)*** \\_sg_qF*$\"3o!4E>!)HJh(F*7$$\"3zkmmO$GdL(F*$\"3%>`9ydD!>')F*7$$\"3&** ****\\_?!QwF*$\"3juG3`J,K(*F*7$$\"3#4L$3x@%>\"zF*$\"3W%oC4vmw2\"F-7$$ \"3x*****\\*3T6#)F*$\"3S.\\N*=bf>\"F-7$$\"3_kmT?w=$\\)F*$\"3(*R'4R7(G6 8F-7$$\"3-++v)[Dxy)F*$\"3D:o'*e%phV\"F-7$$\"33mmm\"4!pv!*F*$\"3Kva'p#e oi:F-7$$\"3Y)**\\PMirP*F*$\"3w')4()et(**p\"F-7$$\"3OMLL`f^n'*F*$\"3eXJ C\"fPq$=F-7$$\"3GKL$eXWW'**F*$\"3W$4p)3,E#)>F-7$$\"3cm;/C9*e-\"F-$\"34 Z7Xub[J@F-7$$\"3++++R,&H0\"F-$\"3/!)o(QI5LF#F-7$$\"3smm\"*zC'R3\"F-$\" 3^\"=9QS`:W#F-7$$\"3ELLL(G+<6\"F-$\"3#4#egyfK(f#F-7$$\"3****\\PvXFT6F- $\"3[8wAB!o!pFF-7$$\"3!***\\iU4ep6F-$\"3'*zBkeX0RHF-7$$\"3%*********** ***>\"F-$\"37++++++GJF--%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!Fb[lFa[l-%*TH ICKNESSG6#\"\"#-%+AXESLABELSG6$Q\"x6\"Q!F[\\l-%%VIEWG6$;$!\"#F`[l$\"#7 F`[l%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "A graph suggest s that there is a zero approximately where " }{XPPEDIT 18 0 "x=0" "6#/ %\"xG\"\"!" }{TEXT -1 5 ".45. " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "f(.45) = -0;" "6#/-%\"fG6#-%&FloatG6$\"#X!\"#,$\"\"!! \"\"" }{TEXT -1 45 ".008875, and f is increasing on the interval " } {XPPEDIT 18 0 "[0,1]" "6#7$\"\"!\"\"\"" }{TEXT -1 15 ", the estimate \+ " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 16 ".45 is too low. \+ " }}{PARA 0 "" 0 "" {TEXT -1 5 "Now " }{XPPEDIT 18 0 "f(.453) = -0;" "6#/-%\"fG6#-%&FloatG6$\"$`%!\"$,$\"\"!!\"\"" }{TEXT -1 15 ".001040323 and " }{XPPEDIT 18 0 "f(0.454)=0" "6#/-%\"fG6#-%&FloatG6$\"$a%!\"$\" \"!" }{TEXT -1 103 ".001576664, so the Intermediate Value Theorem guar antees that there is a zero between 0.453 and 0.454. " }}{PARA 0 "" 0 "" {TEXT -1 8 "In fact " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 15 " has the zero " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" } {TEXT -1 43 ".4533976515, correct to 10 decimal places. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "f := \+ x -> x^3+2*x-1: 'f(x)'=f(x);\nf(.453),f(0.454);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,(*$)F'\"\"$\"\"\"F,*&\"\"#F,F'F,F,F,!\" \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$!(B./\"!\"*$\"(kmd\"F%" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "f := x -> x^3+2*x-1: 'f(x)'=f(x);\nevalf[18](fsolve(f(x)=0));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,(*$)F'\"\"$\"\"\"F,*&\" \"#F,F'F,F,F,!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"3oPS;:l(R`%!# =" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 18 "C ode for pictures " }}{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 0 456 "f := x ->2+(x+2)^3/3: \np1 := plot(f(x),x=0.5..1.5,thickness=2,co lor=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=po int,symbol=[circle,diamond,cross],color=red):\nt1 := plots[textplot]([ [1,-.6,`x = c`],[0.85,12,`(c,f(c))`]],color=COLOR(RGB,.01,0,0)):\nt2 : = plots[textplot]([[1.45,13,`y = f(x)`]],color=red):\nplots[display]([ p||(1..4),t1,t2],view=[0..2,-.6..17],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 45 "Discontinuity at a point in an open int erval " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 851 "g := x ->4+(x-2)^2: \np1 := plot(g(x),x=0.32..0.98,t hickness=2,color=red):\np2 := plot(g(x),x=1.02..1.98,thickness=2,color =red):\np3 := plot([[[.3,0],[.3,g(.3)]],[[2,0],[2,g(2)]]],linestyle=3, color=COLOR(RGB,.3,.3,.3)):\np4 := plot([[[1,0],[1,g(1)],[0,g(1)]]],li nestyle=2,color=COLOR(RGB,.5,.2,.4)):\np5 := plot([[[.3,g(.3)],[1,g(1) ],[2,g(2)]]$2],style=point,symbol=[circle$2],\n symbolsize=[15,18],c olor=red):\np6 := plot(0,x=0.32..1.98,thickness=3,color=COLOR(RGB,0,.8 ,0)):\np7 := plot([[[.3,0],[2,0]]$2],style=point,symbol=[circle$2],\n \+ symbolsize=[15,18],color=COLOR(RGB,0,.8,0)):\nt1 := plots[textplot]( [[2.2,-.2,`x`],[-.05,8,`y`],[-.1,g(1),`L`],\n [.3,-.3,`a`],[1, -.3,`c`],[2,-.3,`b`]],color=COLOR(RGB,.01,0,0)):\nt2 := plots[textplot ]([[1.7,4.8,`y = f(x)`]],color=red):\nplots[display]([p||(1..7),t1,t2] ,view=[-.5..2.2,-0.5..8],tickmarks=[0,0]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1033 "g := x ->4+(x-2) ^2: h := x -> 6.5-0.8*(x-3)^2: \np1 := plot(g(x),x=0.32..0.98,thicknes s=2,color=red):\np2 := plot(h(x),x=1.01..1.98,thickness=2,color=red): \np3 := plot([[[.3,0],[.3,g(.3)]],[[2,0],[2,h(2)]]],linestyle=3,color= COLOR(RGB,.3,.3,.3)):\np4 := plot([[[1,0],[1,g(1)],[0,g(1)]],[[1,h(1)] ,[0,h(1)]]],\n linestyle=2,color=COLOR(RGB,.5,.2,.4)):\np5 \+ := plot([[[.3,g(.3)],[1,g(1)],[2,h(2)]]$2],style=point,symbol=[circle$ 2],\n symbolsize=[15,18],color=red):\np6 := plot(0,x=0.32..1.98,thic kness=3,color=COLOR(RGB,0,.8,0)):\np7 := plot([[[.3,0],[2,0]]$2],style =point,symbol=[circle$2],\n symbolsize=[15,18],color=COLOR(RGB,0,.8, 0)):\np8 := plot([[[1,h(1)]]$4],style=point,symbol=[circle$2,diamond,c ross],\n symbolsize=[15,10$3],color=red):\nt1 := plots[textplot]([[2 .2,-.2,`x`],[-.05,8,`y`],[-.1,g(1),`L`],[-.1,h(1),`R`],\n [.3, -.3,`a`],[1,-.3,`c`],[2,-.3,`b`]],color=COLOR(RGB,.01,0,0)):\nt2 := pl ots[textplot]([[1.5,5.5,`y = f(x)`]],color=red):\nplots[display]([p||( 1..8),t1,t2],view=[-.5..2.2,-0.5..8],tickmarks=[0,0]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1037 "g := \+ x ->6.5-(x-1.6)^2: h := x -> 6.14+sin(4*(1-x)): \np1 := plot(g(x),x=0. 32..0.98,thickness=2,color=red):\np2 := plot(h(x),x=1.01..1.98,thickne ss=2,color=red):\np3 := plot([[[.3,0],[.3,g(.3)]],[[2,0],[2,h(2)]]],li nestyle=3,color=COLOR(RGB,.3,.3,.3)):\np4 := plot([[[1,0],[1,g(1)],[0, g(1)]],[[1,4],[0,4]]],\n linestyle=2,color=COLOR(RGB ,.5,.2,.4)):\np5 := plot([[[.3,g(.3)],[1,g(1)],[2,h(2)]]$2],style=poin t,symbol=[circle$2],\n symbolsize=[15,18],color=red):\np6 := plot(0, x=0.32..1.98,thickness=3,color=COLOR(RGB,0,.8,0)):\np7 := plot([[[.3,0 ],[2,0]]$2],style=point,symbol=[circle$2],\n symbolsize=[15,18],colo r=COLOR(RGB,0,.8,0)):\np8 := plot([[[1,4]]$4],style=point,symbol=[circ le$2,diamond,cross],\n symbolsize=[15,10$3],color=red):\nt1 := plots [textplot]([[2.2,-.2,`x`],[-.05,8,`y`],[-.1,g(1),`L`],[-.1,4,`f(c)`], \n [.3,-.3,`a`],[1,-.3,`c`],[2,-.3,`b`]],color=COLOR(RGB,.01,0 ,0)):\nt2 := plots[textplot]([[1.5,6.5,`y = f(x)`]],color=red):\nplots [display]([p||(1..8),t1,t2],view=[-.5..2.2,-0.5..8],tickmarks=[0,0]); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 259 "" 0 "" {TEXT -1 39 "Code for graph of the tangent function " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 439 "p1 := plot(tan(t*Pi),t=-1.1 ..3.3,-4.9..4.9,discont=true,color=red,thickness=2):\np2 := plot([[[-1 /2,-4.9],[-1/2,4.9]],[[1/2,-4.9],[1/2,4.9]],\n [[3/2,-4.9],[3/2,4.9]], [[5/2,-4.9],[5/2,4.9]]],linestyle=3,color=black):\nt1 := plots[textplo t]([[-.1,4.9,`y`],[3.3,-.3,`x`]],font=[HELVETICA,10]):\nplots[display] ([p1,p2,t1],xtickmarks=[-1=`-p`,-.5=`-p/2`,.5=`p/2`,1=`p`,\n 1.5=`3p /2`,2=`2p`,2.5=`5p/2`,3=`3p`],\n font=[SYMBOL,11],labels=[``,``]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 32 "The intermediate va lue theorem " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1037 "g := x -> .4 + sin(3*x)/3+cos(5*x)/5+x: \np1 : = plot(g(x),x=0.3..2,thickness=2,color=red):\np2 := plot([[[.3,0],[.3, g(.3)],[0,g(.3)]],\n [[2,0],[2,g(2)],[0,g(2)]]],linestyle=3,color=COL OR(RGB,.3,.3,.3)):\np3 := plot([[[1,0],[1,g(1)],[0,g(1)]]],linestyle=2 ,color=COLOR(RGB,.5,.2,.4)):\np4 := plot([[[.3,g(.3)],[1,g(1)],[2,g(2) ]]$4],style=point,symbol=[circle$2,diamond,cross],\n symbolsize=[15, 10$3],color=red):\np5 := plot(0,x=0.32..1.98,thickness=3,color=COLOR(R GB,0,.8,0)):\np6 := plot([[[.3,0],[2,0]]$4],style=point,symbol=[circle $2,diamond,cross],\n symbolsize=[15,10$3],color=COLOR(RGB,0,.8,0)): \np7 := plot([[[1,0],[0,g(.3)],[0,g(1)],[0,g(2)]]$3],style=point,\n \+ symbol=[circle,diamond,cross],color=black):\nt1 := plots[te xtplot]([[2.2,-.1,`x`],[-.05,2.6,`y`],[.3,-.08,`a`],\n [1,-.08,`c`], [2,-.08,`b`],[-.1,g(.3),`f(a)`],[-.15,g(1),`f(c)=k`],[-.1,g(2),`f(b)`] ],\n color=COLOR(RGB,.01,0,0)):\nt2 := plots[textplot]([[1.5,1.8,`y = \+ f(x)`]],color=red):\nplots[display]([p||(1..7),t1,t2],view=[-.15..2.2, -0.1..2.6],tickmarks=[0,0]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK "0 0" 31 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }