﻿
Section2.2
Continuity
63
Sincef iscontinuousatt,thereisanopenintervalI
t
f.x/>
f.t/C˛
2
if x2I
t
\Œa;b
(2.2.8)
(Exercise2.2.15).ThecollectionHD
˚
I
t
ˇ
ˇ
atb
isanopencoveringofŒa;b.Since
Œa;biscompact,theHeine–Boreltheoremimpliesthatthereareﬁnitelymanypointst
1
,
t
2
,...,t
n
suchthattheintervalsI
t
1
,I
t
2
,...,I
t
n
coverŒa;b.Deﬁne
˛
1
D min
1in
f.t
i
/C˛
2
:
Then,sinceŒa;b
S
n
iD1
.I
t
i
\Œa;b/,(2.2.8)impliesthat
f.t/>˛
1
; atb:
But˛
1
1
/D˛forsomex
1
in
Œa;b.
Example2.2.12
Weusedthecompactness ofŒa;bintheproofofTheorem2.2.9
whenweinvokedtheHeine–Boreltheorem. Toseethatcompactnessisessentialtothe
proof,considerthefunction
g.x/D1.1x/sin
1
x
;
whichiscontinuousandhassupremum2onthenoncompactinterval.0;1,butdoesnot
assumeitssupremumon.0;1,since
g.x/1C.1x/
ˇ
ˇ
ˇ
ˇ
sin
1
x
ˇ
ˇ
ˇ
ˇ
1C.1x/<2 if 0<x1:
Asanotherexample,considerthefunction
f.x/De
x
;
whichiscontinuousandhasinﬁmum0,whichitdoesnotattain,onthenoncompactinterval
.0;1/.
Thenexttheoremshowsthatiff iscontinuousonaﬁniteclosedintervalŒa;b,thenf
assumeseveryvaluebetweenf.a/andf.b/asxvariesfromatob(Figure2.2.5,page64).
Theorem2.2.10(Intermediate ValueTheorem)
Supposethatf iscon-
tinuousonŒa;b;f.a/ ¤ f.b/;andisbetweenf.a/andf.b/:Thenf.c/ D for
somecin.a;b/:
Pdf hyperlink - insert, remove PDF links in C#.net, ASP.NET, MVC, Ajax, WinForms, WPF
Free C# example code is offered for users to edit PDF document hyperlink (url), like inserting and deleting
Pdf hyperlink - VB.NET PDF url edit library: insert, remove PDF links in vb.net, ASP.NET, MVC, Ajax, WinForms, WPF
Help to Insert a Hyperlink to Specified PDF Document Page
64 Chapter2
DifferentialCalculusofFunctionsofOneVariable
a
b
x
x
y
y = f(x)
y = µ
Figure2.2.5
Proof
Supposethatf.a/<<f.b/.Theset
SD
˚
x
ˇ
ˇ
axb and f.x/
isboundedandnonempty. LetcD supS. . Wewillshowthatf.c/D D . . Iff.c/> > ,
thenc > > aand,sincef iscontinuousatc, , thereisan > > 0suchthatf.x/ > > if
c < x   c c (Exercise 2.2.15). Therefore, , cisanupperboundforS, which
an>0suchthatf.x/<forcx<cC,socisnotanupperboundforS.Thisis
Theproofforthecasewheref.b/<<f.a/canbeobtainedbyapplyingthisresult
tof.
UniformContinuity
Theorem2.2.2andDeﬁnition2.2.3implythata
functionf iscontinuousonasubsetSofitsdomainifforeach>0andeachx
0
inS,
thereisaı>0,whichmaydependuponx
0
aswellas,suchthat
jf.x/f.x
0
/j< if jxx
0
j<ı and
x2D
f
:
ThenextdeﬁnitionintroducesanotherkindofcontinuityonasetS.
Deﬁnition2.2.11
Afunctionf isuniformlycontinuousonasubsetS S ofitsdomain
if,forevery>0,thereisaı>0suchthat
jf.x/f.x
0
/j<whenever jxx
0
j<ıand x;x
0
2S:
WeemphasizethatinthisdeﬁnitionıdependsonlyonandSandnotontheparticular
choiceofxandx0,providedthattheyarebothinS.
Example2.2.13
Thefunction
f.x/D2x
How to C#: Basic SDK Concept of XDoc.PDF for .NET
XDoc.PDF for .NET allows C# developers to edit hyperlink of PDF document, including editing PDF url links and quick navigation link in bookmark/outline.
VB.NET PDF: Basic SDK Concept of XDoc.PDF
XDoc.PDF for .NET allows VB.NET developers to edit hyperlink of PDF document, including editing PDF url links and quick navigation link in bookmark/outline.
Section2.2
Continuity
65
isuniformlycontinuouson.1;1/,since
jf.x/f.x
0
/jD2jxx
0
j< if jxx
0
j<=2:
Example2.2.14
If0<r<1,thenthefunction
g.x/Dx
2
isuniformlycontinuousonŒr;r.Toseethis,notethat
jg.x/g.x
0
/Djx
2
.x
0
/
2
jDjxx
0
jjxCx
0
j2rjxx
0
j;
so
jg.x/g.x
0
/j< if
jxx
0
j<ıD
2r
and rx;x
0
r:
Oftenaconceptisclariﬁedbyconsideringitsnegation:afunctionf isnotuniformly
continuousonSifthereisan
0
>0suchthatifıisanypositivenumber,therearepoints
xandx
0
inSsuchthat
jxx
0
j<ı but jf.x/f.x
0
/j
0
:
Example2.2.15
Thefunctiong.x/Dx
2
isuniformlycontinuousonŒr;rforany
ﬁniter(Example2.2.14),butnoton.1;1/. Toseethis,wewillshowthatifı ı >0
therearerealnumbersxandx
0
suchthat
jxx
0
jDı=2 and
jg.x/g.x
0
/j1:
Tothisend,wewrite
jg.x/g.x
0
/jDjx
2
.x
0
/
2
jDjxx
0
jjxCx
0
j:
Ifjxx0jDı=2andx;x>1=ı,then
jxx
0
jjxCx
0
j>
ı
2
1
ı
C
1
ı
D1:
Example2.2.16
Thefunction
f.x/Dcos
1
x
iscontinuouson.0;1(Exercise2.2.23
(i)
). However,f isnotuniformlycontinuouson
.0;1,since
ˇ
ˇ
ˇ
ˇ
f
1
n
f
1
.nC1/
ˇ
ˇ
ˇ
ˇ
D2; nD1;2;::::
Examples2.2.15and2.2.16showthatafunctionmaybecontinuousbutnotuniformly
continuousonaninterval. Thenexttheoremshowsthatthiscannothappeniftheinterval
isclosedandbounded,andthereforecompact.
VB.NET Create PDF from Word Library to convert docx, doc to PDF in
Ability to get word count of PDF pages. Change Word hyperlink to PDF hyperlink and bookmark. Free online Word to PDF converter without email.
VB.NET Create PDF from Excel Library to convert xlsx, xls to PDF
Merge all Excel sheets to one PDF file in VB.NET. Change Excel hyperlink to PDF hyperlink and bookmark. Export PDF from Excel with cell border or no border.
66 Chapter2
DifferentialCalculusofFunctionsofOneVariable
Theorem2.2.12
Iff iscontinuousonaclosedandboundedintervalŒa;b;thenf
isuniformlycontinuousonŒa;b:
Proof
Supposethat>0.Sincef iscontinuousonŒa;b,foreachtinŒa;bthereis
apositivenumberı
t
suchthat
jf.x/f.t/j<
2
if jxtj<2ı
t
and x2Œa;b:
(2.2.9)
IfI
t
D.tı
t
;tCı
t
/,thecollection
HD
˚
I
t
ˇ
ˇ
t2Œa;b
isanopencoveringofŒa;b.SinceŒa;biscompact,theHeine–Boreltheoremimpliesthat
thereareﬁnitelymanypointst
1
,t
2
,...,t
n
inŒa;bsuchthatI
t
1
,I
t
2
,...,I
t
n
coverŒa;b.
Nowdeﬁne
ıDminfı
t
1
t
2
;:::;ı
t
n
g:
(2.2.10)
Wewillshowthatif
jxx
0
j<ı and x;x
0
2Œa;b;
(2.2.11)
thenjf.x/f.x
0
/j<.
Fromthetriangleinequality,
jf.x/f.x0/jDj.f.x/f.t
r
//C.f.t
r
/f.x0//j
jf.x/f.t
r
/jCjf.t
r
/f.x0/j:
(2.2.12)
SinceI
t
1
,I
t
2
,...,I
t
n
coverŒa;b,xmustbeinoneoftheseintervals.Supposethatx2I
t
r
;
thatis,
jxt
r
j<ı
t
r
:
(2.2.13)
From(2.2.9)withtDt
r
,
jf.x/f.t
r
/j<
2
:
(2.2.14)
From(2.2.11),(2.2.13),andthetriangleinquality,
jx
0
t
r
jDj.x
0
x/C.xt
r
/jjx
0
xjCjxt
r
j<ıCı
t
r
2ı
t
r
:
Therefore,(2.2.9)withtDt
r
andxreplacedbyx
0
impliesthat
jf.x
0
/f.t
r
/j<
2
:
This,(2.2.12),and(2.2.14)implythatjf.x/f.x
0
/j<.
ThisproofagainshowstheutilityoftheHeine–Boreltheorem,whichallowedusto
deﬁneıin(2.2.10)asthesmallestofaﬁnitesetofpositivenumbers,sothatıissuretobe
positive.(Aninﬁnitesetofpositivenumbersmayfailtohaveasmallestpositivemember;
forexample,considertheopeninterval.0;1/.)
Corollary2.2.13
Iff iscontinuousonasetT;thenf isuniformlycontinuouson
anyﬁniteclosedintervalcontainedinT:
VB.NET PDF Library SDK to view, edit, convert, process PDF file
Please click to see details. PDF Hyperlink Edit. RasterEdge PDF SDK for .NET package offers robust APIs for editing PDF document
C# PDF Library SDK to view, edit, convert, process PDF file for C#
Please click to see details. C#.NET: Edit PDF Hyperlink. RasterEdge PDF SDK for .NET package offers robust APIs for editing PDF document
Section2.2
Continuity
67
AppliedtoExample2.2.16,Corollary2.2.13impliesthatthefunctiong.x/Dcos1=x
isuniformlycontinuousonŒ;1if0<<1.
Theorem2.1.9impliesthatiff ismonotoniconanintervalI,thenf iseithercontinuous
orhasajumpdiscontinuityateachx
0
inI. ThisandTheorem2.2.10providethekeyto
theproofofthefollowingtheorem.
Theorem2.2.14
IffismonotonicandnonconstantonŒa;b;thenfiscontinuouson
Œa;bifandonlyifitsrangeR
f
D
˚
f.x/
ˇ
ˇ
x2Œa;b
istheclosedintervalwithendpoints
f.a/andf.b/:
Proof
Weassumethatf isnondecreasing,andleavethecasewheref isnonincreasing
toyou(Exercise2.2.34).Theorem2.1.9
(a)
impliesthattheset
e
R
f
D
˚
f.x/
ˇ
ˇ
x2.a;b/
isasubsetoftheopeninterval.f.aC/;f.b//.Therefore,
R
f
Dff.a/g[
e
R
f
[ff.b/gff.a/g[.f.aC/;f.b//[ff.b/g:
(2.2.15)
Nowsupposethatf iscontinuousonŒa;b. . Thenf.a/D D f.aC/,f.b/ D f.b/,so
(2.2.15)impliesthatR
f
 Œf.a/;f.b/. Iff.a/ < <   < < f.b/,thenTheorem 2.2.10
impliesthatDf.x/forsomexin.a;b/.Hence,R
f
DŒf.a/;f.b/.
Fortheconverse,supposethatR
f
DŒf.a/;f.b/.Sincef.a/f.aC/andf.b/
f.b/,(2.2.15)impliesthatf.a/ D f.aC/andf.b/ Df.b/. WeknowfromTheo-
rem2.1.9
(c)
thatiff isnondecreasinganda<x
0
<b,then
f.x
0
/f.x
0
/f.x
0
C/:
Ifeitheroftheseinequalitiesisstrict,R
f
ourassumption,f.x
0
/D f.x
0
/D f.x
0
C/. Therefore,f iscontinuousatx
0
(Exer-
cise2.2.2).Wecannowconcludethatf iscontinuousonŒa;b.
Theorem2.2.14impliesthefollowingtheorem.
Theorem2.2.15
SupposethatfisincreasingandcontinuousonŒa;b;andletf.a/D
candf.b/Dd:ThenthereisauniquefunctiongdeﬁnedonŒc;dsuchthat
g.f.x//Dx; axb;
(2.2.16)
and
f.g.y//Dy; cyd:
(2.2.17)
Moreover;giscontinuousandincreasingonŒc;d:
Proof
Weﬁrstshowthatthereisafunctiongsatisfying(2.2.16)and(2.2.17).Sincef
iscontinuous,Theorem2.2.14impliesthatforeachy
0
inŒc;dthereisanx
0
inŒa;bsuch
that
f.x
0
/Dy
0
;
(2.2.18)
C# Create PDF from Word Library to convert docx, doc to PDF in C#.
Able to get word count in PDF pages. Change Word hyperlink to PDF hyperlink and bookmark. Free online Word to PDF converter without email.
.NET PDF SDK - Description of All PDF Processing Control Feastures
68 Chapter2
DifferentialCalculusofFunctionsofOneVariable
and,sincef isincreasing,thereisonlyonesuchx
0
.Deﬁne
g.y
0
/Dx
0
:
(2.2.19)
Thedeﬁnitionofx
0
isillustratedinFigure2.2.6:withŒc;ddrawnonthey-axis,ﬁndthe
intersectionoftheliney D y
0
withthecurvey D D f.x/anddropaverticalfromthe
intersectiontothex-axistoﬁndx
0
.
y
d
c
a
b
x
y = f(x)
x
0
y
0
Figure2.2.6
Substituting(2.2.19)into(2.2.18)yields
f.g.y
0
//Dy
0
;
andsubstituting(2.2.18)into(2.2.19)yields
g.f.x
0
//Dx
0
:
Droppingthesubscriptsinthesetwoequationsyields(2.2.16)and(2.2.17).
Theuniquenessofgfollowsfromourassumptionthatf isincreasing,andtherefore
onlyonevalueofx
0
cansatisfy(2.2.18)foreachy
0
.
Toseethatgisincreasing,supposethaty
1
<y
2
andletx
1
andx
2
bethepointsinŒa;b
suchthatf.x
1
/Dy
1
andf.x
2
/Dy
2
.Sincefisincreasing,x
1
<x
2
.Therefore,
g.y
1
/Dx
1
<x
2
Dg.y
2
/;
sogisincreasing. SinceR
g
D
˚
g.y/
ˇ
ˇ
y2Œc;d
istheintervalŒg.c/;g.d/ D Œa;b,
Theorem2.2.14withf andŒa;breplacedbygandŒc;dimpliesthatgiscontinuouson
Œc;d.
ThefunctiongofTheorem2.2.15istheinverseoff,denotedbyf1.Since(2.2.16)
and(2.2.17)aresymmetricinf andg,wecanalsoregardfastheinverseofg,anddenote
itbyg1.
Section2.2
Continuity
69
Example2.2.17
If
f.x/Dx
2
; 0xR;
then
f
1
.y/Dg.y/D
p
y; 0yR
2
:
Example2.2.18
If
f.x/D2xC4; 0x2;
then
f
1
.y/Dg.y/D
y4
2
; 4y8:
2.2Exercises
1.
ProveTheorem2.2.2.
2.
Provethatafunctionf iscontinuousatx
0
ifandonlyif
lim
x!x
0
f.x/D
lim
x!x
0
C
f.x/Df.x
0
/:
3.
Determinewhetherfiscontinuousordiscontinuousfromtherightorleftatx
0
.
(a)
f.x/D
p
x .x
0
D0/
(b)
f.x/D
p
x .x
0
>0/
(c)
f.x/D
1
x
.x
0
D0/
(d)
f.x/Dx
2
.x
0
arbitrary/
(e)
f.x/D
xsin1=x; x¤0;
1;
xD0
.x
0
D0/
(f)
f.x/D
xsin1=x; x¤0
0;
xD0
.x
0
D0/
(g)
f.x/D
8
<
:
xCjxj.1Cx/
x
sin
1
x
; x¤0
1;
xD0
.x
0
D0/
4.
Letf bedeﬁnedonŒ0;2by
f.x/D
(
x
2
;
0x<1;
xC1; 1x2:
Onwhichofthefollowingintervalsisf continuousaccordingtoDeﬁnition2.2.3:
Œ0;1/,.0;1/,.0;1,Œ0;1,Œ1;2/,.1;2/,.1;2,Œ1;2?
5.
Let
g.x/D
p
x
x1
:
OnwhichofthefollowingintervalsisgcontinuousaccordingtoDeﬁnition2.2.3:
Œ0;1/,.0;1/,.0;1,Œ1;1/,.1;1/?
70 Chapter2
DifferentialCalculusofFunctionsofOneVariable
6.
Let
f.x/D
(
-1 ifxisirrational;
1 ifxisrational:
Showthatf isnotcontinuousanywhere.
7.
Letf.x/ D 0ifxisirrationalandf.p=q/ D 1=qifpandqarepositiveinte-
gerswithnocommonfactors. Showthatf isdiscontinuousateveryrationaland
continuousateveryirrationalon.0;1/.
8.
Prove:Iff assumesonlyﬁnitelymanyvalues,thenf iscontinuousatapointx
0
in
D
0
f
ifandonlyiff isconstantonsomeinterval.x
0
ı;x
0
Cı/.
9.
Thecharacteristicfunction
T
ofasetTisdeﬁnedby
T
.x/D
(
1; x2T;
0; x62T:
Showthat
T
iscontinuousatapointx
0
ifandonlyifx
0
2T
0
[.T
c
/
0
.
10.
subset(Deﬁnition1.1.5)of.a;b/,thenf.x/Dg.x/forallxin.a;b/.
11.
Provethatthefunctiong.x/Dlogxiscontinuouson.0;1/. Takethefollowing
propertiesasgiven.
(a)
lim
x!1
g.x/D0.
(b)
g.x
1
/Cg.x
2
/Dg.x
1
x
2
/ifx
1
;x
2
>0.
12.
Provethatthefunctionf.x/Deax iscontinuouson.1;1/.Takethefollowing
propertiesasgiven.
(a)
lim
x!0
f.x/D1.
(b)
f.x
1
Cx
2
/Df.x
1
/f.x
2
/; 1<x
1
;x
2
<1.
13. (a)
Provethatthefunctionssinhxandcoshxarecontinuousforallx.
(b)
Forwhatvaluesofxaretanhxandcothxcontinuous?
14.
Provethatthefunctionss.x/Dsinxandc.x/Dcosxarecontinuouson.1;1/.
Takethefollowingpropertiesasgiven.
(a)
lim
x!0
c.x/D1.
(b)
c.x
1
x
2
/Dc.x
1
/c.x
2
/Cs.x
1
/s.x
2
/; 1<x
1
;x
2
<1.
(c)
s
2
.x/Cc
2
.x/D1; 1<x<1.
15. (a)
Prove:Iff iscontinuousatx
0
andf.x
0
/>,thenf.x/>forallxin
someneighborhoodofx
0
.
(b)
Statearesultanalogousto
(a)
forthecasewheref.x
0
/<.
(c)
Prove:Iff.x/forallxinSandx
0
isalimitpointofSatwhichf is
continuous,thenf.x
0
/.
(d)
Stateresultsanalogousto
(a)
,
(b)
,and
(c)
forthecasewheref iscontin-
uousfromtherightorleftatx
0
.
Section2.2
Continuity
71
16.
Letjfjbethefunctionwhosevalueateachx inD
f
isjf.x/j. Prove: Iff is
continuousatx
0
,thensoisjfj.Istheconversetrue?
17.
Prove:Iff ismonotoniconŒa;b,thenf ispiecewisecontinuousonŒa;bifand
onlyiff hasonlyﬁnitelymanydiscontinuitiesinŒa;b.
18.
ProveTheorem2.2.5.
19. (a)
Showthatiff
1
,f
2
,...,f
n
arecontinuousonasetSthensoaref
1
Cf
2
C
Cf
n
andf
1
f
2
f
n
.
(b)
Use
(a)
toshowthatarationalfunctioniscontinuousforallvaluesofx
exceptthezerosofitsdenominator.
20. (a)
Letf
1
andf
2
becontinuousatx
0
anddeﬁne
F.x/Dmax.f
1
.x/;f
2
.x//:
ShowthatFiscontinuousatx
0
.
(b)
Letf
1
,f
2
,...,f
n
becontinuousatx
0
anddeﬁne
F.x/Dmax.f
1
.x/;f
2
.x/;:::;f
n
.x//:
ShowthatFiscontinuousatx
0
.
21.
Findthedomainsoffıgandgıf.
(a)
f.x/D
p
x; g.x/D1x
2
(b)
f.x/Dlogx; g.x/Dsinx
(c)
f.x/D
1
1x2
; g.x/Dcosx
(d)
f.x/D
p
x; g.x/Dsin2x
22. (a)
Supposethaty
0
D lim
x!x
0
g.x/existsandisaninteriorpointofD
f
,and
thatf iscontinuousaty
0
.Showthat
lim
x!x
0
.f ıg/.x/Df.y
0
/:
(b)
Stateananalogousresultforlimitsfromtheright.
(c)
Stateananalogousresultforlimitsfromtheleft.
23.
UseTheorem2.2.7toﬁndallpointsx
0
atwhichthefollowingfunctionsarecontin-
uous.
(a)
p
1x2
(b)
sinex
2
(c)
log.1Csinx/
(d)
e1=.12x/
(e)
sin
1
.x1/2
(f)
sin
1
cosx
(g)
.1sin
2
x/
1=2
(h)
cot.1e
x
2
/
(i)
cos
1
x
24.
CompletetheproofofTheorem 2.2.9byshowingthatthereisan n x
2
suchthat
f.x
2
/ D ˇ.