Section2.2
Continuity
63
Sincef iscontinuousatt,thereisanopenintervalI
t
abouttsuchthat
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
>˛,sothiscontradictsthedefinitionof˛.Therefore,f.x
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/:
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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
contradictsthedefinitionofcasthesupremumofS.Iff.c/<,thenc<bandthereis
an>0suchthatf.x/<forcx<cC,socisnotanupperboundforS.Thisis
alsoacontradiction.Therefore,f.c/D.
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
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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.
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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:
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Section2.2
Continuity
67
AppliedtoExample2.2.16,Corollary2.2.13impliesthatthefunctiong.x/Dcos1=x
isuniformlycontinuousonŒ;1if0<<1.
MoreAbout MonotonicFunctions
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
cannotbeaninterval. Sincethiscontradicts
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)
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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.
Prove:Iff andgarecontinuouson.a;b/andf.x/Dg.x/foreveryxinadense
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.
72 Chapter2
DifferentialCalculusofFunctionsofOneVariable
25.
Prove: Iff isnonconstantandcontinuous s onanintervalI, thenthesetS D
˚
y
ˇ
ˇ
yDf.x/;x2I
isaninterval.Moreover,ifI isafiniteclosedinterval,then
soisS.
26.
Supposethatf andgaredefinedon.1;1/,f isincreasing,andfıgiscon-
tinuouson.1;1/.Showthatgiscontinuouson.1;1/.
27.
LetfbecontinuousonŒa;b/,anddefine
F.x/D max
atx
f.t/; ax<b:
(HowdoweknowthatFiswelldefined?)ShowthatF iscontinuousonŒa;b/.
28.
LetfandgbeuniformlycontinuousonanintervalS.
(a)
Showthatf CgandfgareuniformlycontinuousonS.
(b)
ShowthatfgisuniformlycontinuousonSifSiscompact.
(c)
Showthatf=gisuniformlycontinuousonS ifS iscompactandghasno
zerosinS.
(d)
Giveexamplesshowingthattheconclusionof
(b)
and
(c)
mayfailtohold
ifSisnotcompact.
(e)
Stateadditionalconditionsonf andgwhichguaranteethatfgisuniformly
continuousonSevenifSisnotcompact.Dothesameforf=g.
29.
Supposethatf isuniformlycontinuousonasetS,gisuniformlycontinuousona
setT,andg.x/2SforeveryxinT.Showthatf ıgisuniformlycontinuouson
T.
30. (a)
Prove: Iff isuniformlycontinuousondisjointclosedintervalsI
1
,I
2
,...,
I
n
,thenf isuniformlycontinuouson
S
n
jD1
I
j
.
(b)
Is
(a)
validwithouttheword“closed”?
31. (a)
Prove:Iff isuniformlycontinuousonaboundedopeninterval.a;b/,then
f.aC/andf.b/existandarefinite.H
INT
:SeeExercise2.1.38:
(b)
Showthattheconclusionin
(a)
doesnotfollowif.a;b/isunbounded.
32.
Prove:Iff iscontinuousonŒa;1/andf.1/exists(finite),thenf isuniformly
continuousonŒa;1/.
33.
Supposethatf isdefinedon.1;1/andhasthefollowingproperties.
(i)
lim
x!0
f.x/D1and
(ii)
f.x
1
Cx
2
/Df.x
1
/f.x
2
/; 1<x
1
;x
2
<1:
Prove:
(a)
f.x/>0forallx.
(b)
f.rx/DŒf.x/
r
ifrisrational.
(c)
Iff.1/D1thenf isconstant.
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