﻿
Section1.3
TheRealLine
23
Example1.3.5
If1<a<b<1,theset
Œa;bD
˚
x
ˇ
ˇ
axb
isclosed,sinceitscomplementistheunionoftheopensets.1;a/and.b;1/.Wesay
thatŒa;bisaclosedinterval.Theset
Œa;b/D
˚
x
ˇ
ˇ
ax<b
isahalf-closedorhalf-openintervalif1<a<b<1,asis
.a;bD
˚
x
ˇ
ˇ
a<xb
I
however,neitherofthesesetsisopenorclosed.(Whynot?)Semi-inﬁniteclosedintervals
aresetsoftheform
Œa;1/D
˚
x
ˇ
ˇ
ax
and .1;aD
˚
x
ˇ
ˇ
xa
;
wherea isﬁnite. Theyareclosedsets, , sincetheircomplementsaretheopenintervals
.1;a/and.a;1/,respectively.
Example1.3.4showsthatasetmaybebothopenandclosed,andExample1.3.5shows
thatasetmaybeneither.Thus,openandclosedarenotoppositesinthiscontext,asthey
areineverydayspeech.
Example1.3.6
FromTheorem1.3.3andExample1.3.4,theunionofanycollectionof
openintervalsisanopenset.(Infact,itcanbeshownthateverynonemptyopensubsetof
Ristheunionofopenintervals.)FromTheorem1.3.3andExample1.3.5,theintersection
ofanycollectionofclosedintervalsisclosed.
Itcanbeshownthattheintersectionofﬁnitelymanyopensetsisopen, andthatthe
unionofﬁnitelymanyclosedsetsisclosed. However,theintersectionofinﬁnitelymany
opensetsneednotbeopen,andtheunionofinﬁnitelymanyclosedsetsneednotbeclosed
(Exercises1.3.8and1.3.9).
Deﬁnition1.3.4
LetSbeasubsetofR.Then
(a)
x
0
isalimitpointofSifeverydeletedneighborhoodofx
0
containsapointofS.
(b)
x
0
isaboundarypointofSifeveryneighborhoodofx
0
containsatleastonepoint
inSandonenotinS.ThesetofboundarypointsofSistheboundaryofS,denoted
by@S.TheclosureofS,denotedby
S,is
SDS[@S.
(c)
x
0
isanisolatedpointofSifx
0
2Sandthereisaneighborhoodofx
0
thatcontains
nootherpointofS.
(d)
x
0
isexteriortoSifx
0
isintheinteriorofS
c
.Thecollectionofsuchpointsisthe
exteriorofS.
Example1.3.7
LetSD.1;1[.1;2/[f3g.Then
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24 Chapter1
TheRealNumbers
(a)
ThesetoflimitpointsofSis.1;1[Œ1;2.
(b)
@SDf1;1;2;3gand
S D.1;1[Œ1;2[f3g.
(c)
3istheonlyisolatedpointofS.
(d)
TheexteriorofSis.1;1/[.2;3/[.3;1/.
Example1.3.8
Forn1,let
I
n
D
1
2nC1
;
1
2n
and SD
1
[
nD1
I
n
:
Then
(a)
ThesetoflimitpointsofSisS[f0g.
(b)
@SD
˚
x
ˇ
ˇ
xD0orxD1=n.n2/
and
SDS[f0g.
(c)
Shasnoisolatedpoints.
(d)
TheexteriorofSis
.1;0/[
"
[1
nD1
1
2nC2
;
1
2nC1
#
[
1
2
;1
:
Example1.3.9
LetSbethesetofrationalnumbers. Sinceeveryintervalcontainsa
rationalnumber(Theorem1.1.6),everyrealnumberisalimitpointofS;thus,
S DR.
Sinceeveryintervalalsocontainsanirrationalnumber(Theorem1.1.7),everyrealnumber
isaboundarypointofS;thus@SDR.TheinteriorandexteriorofSarebothempty,and
Shasnoisolatedpoints.Sisneitheropennorclosed.
ThenexttheoremsaysthatSisclosedifandonlyifSD
S(Exercise1.3.14).
Theorem1.3.5
AsetSisclosedifandonlyifnopointofS
c
isalimitpointofS:
Proof
SupposethatSisclosedandx
0
2S
c
.SinceS
c
isopen,thereisaneighborhood
ofx
0
thatiscontainedinS
c
andthereforecontainsnopointsofS.Hence,x
0
cannotbea
limitpointofS.Fortheconverse,ifnopointofS
c
isalimitpointofStheneverypointin
S
c
musthaveaneighborhoodcontainedinS
c
.Therefore,S
c
isopenandSisclosed.
Theorem1.3.5isusuallystatedasfollows.
Corollary1.3.6
Asetisclosedifandonlyifitcontainsallitslimitpoints:
Theorem1.3.5andCorollary1.3.6areequivalent. However,westatedthetheoremas
wedidbecausestudentssometimesincorrectlyconcludefromthecorollarythataclosed
setmusthavelimitpoints. Thecorollarydoesnotsaythis. IfShasnolimitpoints,then
thesetoflimitpointsisemptyandthereforecontainedinS. Hence,asetwithnolimit
pointsisclosedaccordingtothecorollary,inagreementwithTheorem1.3.5.Forexample,
anyﬁnitesetisclosed.Moregenerally,Sisclosedifthereisaı>0suchjxyjıfor
everypairfx;ygofdistinctpointsinS.
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Section1.3
TheRealLine
25
OpenCoverings
AcollectionHofopensetsisanopencoveringofasetSifeverypointinSiscontained
inasetHbelongingtoH;thatis,ifS[
˚
H
ˇ
ˇ
H 2H
.
Example1.3.10
Thesets
S
1
DŒ0;1;S
2
Df1;2;:::;n;:::g;
S
3
D
1;
1
2
;:::;
1
n
;:::
; and S
4
D.0;1/
arecoveredbythefamiliesofopenintervals
H
1
D

x
1
N
;xC
1
N
ˇ
ˇ
ˇ
ˇ
0<x<1
; (NDpositiveinteger),
H
2
D

n
1
4
;nC
1
4
ˇ
ˇ
ˇ
ˇ
nD1;2;:::
;
H
3
D
1
nC
1
2
;
1
n
1
2
!
ˇ
ˇ
ˇ
ˇ
nD1;2;:::
)
;
and
H
4
Df.0;/j0<<1g;
respectively.
Theorem1.3.7(HeineBorelTheorem)
IfHisanopencoveringofaclosed
andboundedsubsetSoftherealline;thenShasanopencovering
e
Hconsistingofﬁnitely
manyopensetsbelongingtoH:
Proof
SinceS isbounded,ithasaninﬁmum˛andasupremumˇ,and,sinceS is
closed,˛andˇbelongtoS(Exercise1.3.17).Deﬁne
S
t
DS\Œ˛;t for r t˛;
andlet
F D
˚
t
ˇ
ˇ
˛tˇandﬁnitelymanysetsfromHcoverS
t
:
SinceS
ˇ
DS,thetheoremwillbeprovedifwecanshowthatˇ2F.Todothis,weuse
thecompletenessofthereals.
Since˛2S,S
˛
isthesingletonsetf˛g,whichiscontainedinsomeopensetH
˛
from
ithasasupremum.First,wewishtoshowthatDˇ.SinceˇbydeﬁnitionofF,
itsufﬁcestoruleoutthepossibilitythat<ˇ.Weconsidertwocases.
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26 Chapter1
TheRealNumbers
C
ASE
1.Supposethat<ˇand62S.Then,sinceSisclosed,isnotalimitpoint
ofS(Theorem1.3.5).Consequently,thereisan>0suchthat
Œ;C\SD;;
soS

DS
C
. However,thedeﬁnitionofimpliesthatS

hasaﬁnitesubcovering
fromH,whileS
C
C
ASE
2. Supposethat < < ˇ and 2 2 S. ThenthereisanopensetH
inH that
containsand,alongwith,anintervalŒ;Cforsomepositive.SinceS

has
aﬁnitecoveringfH
1
;:::;H
n
gofsetsfromH,itfollowsthatS
C
hastheﬁnitecovering
fH
1
;:::;H
n
;H
NowweknowthatDˇ,whichisinS.Therefore,thereisanopensetH
ˇ
inHthat
containsˇandalongwithˇ,anintervaloftheformŒˇ;ˇC,forsomepositive.
SinceS
ˇ
iscoveredbyaﬁnitecollectionofsetsfH
1
;:::;H
k
g,S
ˇ
iscoveredbythe
ﬁnitecollectionfH
1
;:::;H
k
;H
ˇ
g.SinceS
ˇ
DS,weareﬁnished.
Henceforth,wewillsaythataclosedandboundedsetiscompact. TheHeine–Borel
theoremsaysthatanyopencoveringofacompactsetScontainsaﬁnitecollectionthat
aswelldeﬁneasetSofrealstobecompactifithastheHeine–Borelproperty;thatis,if
everyopencoveringofScontainsaﬁnitesubcovering.ThesameistrueofRn,whichwe
studyinSection5.1.Thisdeﬁnitiongeneralizestomoreabstractspaces(calledtopological
spaces)forwhichtheconceptofboundednessneednotbedeﬁned.
Example1.3.11
SinceS
1
inExample1.3.10iscompact,theHeine–Boreltheorem
impliesthatS
1
canbecoveredbyaﬁnitenumberofintervalsfromH
1
.Thisiseasilyveri-
ﬁed,since,forexample,the2N intervalsfromH
1
centeredatthepointsx
k
Dk=2N.0
k2N1/coverS
1
.
TheHeine–BoreltheoremdoesnotapplytotheothersetsinExample1.3.10sincethey
arenotcompact:S
2
isunboundedandS
3
andS
4
arenotclosed,sincetheydonotcontain
alltheirlimitpoints(Corollary1.3.6). TheconclusionoftheHeine–Boreltheoremdoes
notholdforthesesetsandtheopencoveringsthatwehavegivenforthem. Eachpointin
S
2
iscontainedinexactlyonesetfromH
2
,soremovingevenoneofthesesetsleavesa
pointofS
2
uncovered.If
e
H
3
isanyﬁnitecollectionofsetsfromH
3
,then
1
n
62[
˚
H
ˇ
ˇ
H 2
e
H
3
fornsufﬁcientlylarge.Anyﬁnitecollectionf.0;
1
/;:::;.0;
n
/gfromH
4
coversonlythe
interval.0;
max
/,where
max
Dmaxf
1
;:::;
n
g<1:
TheBolzano–WeierstrassTheorem
AsanapplicationoftheHeine–Boreltheorem,weprovethefollowingtheoremofBolzano
andWeierstrass.
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Section1.3
TheRealLine
27
Theorem1.3.8(BolzanoWeierstrassTheorem)
Everyboundedinﬁniteset
ofrealnumbershasatleastonelimitpoint:
Proof
Wewillshowthataboundednonemptysetwithoutalimitpointcancontainonly
aﬁnitenumberofpoints. IfS hasnolimitpoints,thenSisclosed(Theorem1.3.5)and
everypointxofShasanopenneighborhoodN
x
thatcontainsnopointofSotherthanx.
Thecollection
HD
˚
N
x
ˇ
ˇ
x2S
isanopencoveringforS. SinceSisalsobounded,Theorem1.3.7impliesthatScanbe
coveredbyaﬁnitecollectionofsetsfromH,sayN
x
1
,...,N
x
n
. Sincethesesetscontain
onlyx
1
,...,x
n
fromS,itfollowsthatSDfx
1
;:::;x
n
g.
1.3Exercises
1.
FindS\T,.S\T/
c
,S
c
\T
c
,S[T,.S[T/
c
,andS
c
[T
c
.
(a)
SD.0;1/,T D
1
2
;
3
2
(b)
SD
˚
x
ˇ
ˇ
x
2
>4
,TD
˚
x
ˇ
ˇ
x
2
<9
(c)
SD.1;1/,T D;
(d)
SD.1;1/,T D.1;1/
2.
LetS
k
D.11=k;2C1=k,k1.Find
(a)
[1
kD1
S
k
(b)
\1
kD1
S
k
(c)
[1
kD1
S
c
k
(d)
\1
kD1
S
c
k
3.
Prove: IfAandBaresetsandthereisasetX suchthatA[X X D D B[X and
4.
FindthelargestsuchthatScontainsan-neighborhoodofx
0
.
(a)
x
0
D
3
4
,SD
1
2
;1
(b)
x
0
D
2
3
,SD
1
2
;
3
2
(c)
x
0
D5,SD.1;1/
(d)
x
0
D1,SD.0;2/
5.
Describe thefollowingsetsasopen, closed,orneither, andﬁndS
0
, .S
c
/
0
, and
.S
0
/
c
.
(a)
SD.1;2/[Œ3;1/
(b)
SD.1;1/[.2;1/
(c)
SDŒ3;2[Œ7;8
(d)
SD
˚
x
ˇ
ˇ
xDinteger
6.
Provethat.S\T/
c
DS
c
[T
c
and.S[T/
c
DS
c
\T
c
.
7.
LetF beacollectionofsetsanddeﬁne
I D\
˚
F
ˇ
ˇ
F 2F
and
U D[
˚
F
ˇ
ˇ
F 2F
:
Provethat
(a)
I
c
D[
˚
F
c
ˇ
ˇ
F 2F
and
(b)
U
c
D
˚
\F
c
ˇ
ˇ
F2F
.
8. (a)
Showthattheintersectionofﬁnitelymanyopensetsisopen.
28 Chapter1
TheRealNumbers
(b)
Giveanexampleshowingthattheintersectionofinﬁnitelymanyopensets
mayfailtobeopen.
9. (a)
Showthattheunionofﬁnitelymanyclosedsetsisclosed.
(b)
Giveanexampleshowingthattheunionofinﬁnitelymanyclosedsetsmay
failtobeclosed.
10.
Prove:
(a)
IfU isaneighborhoodofx
0
andU V,thenV isaneighborhoodofx
0
.
(b)
IfU
1
,...,U
n
areneighborhoodsofx
0
,sois
T
n
iD1
U
i
.
11.
FindthesetoflimitpointsofS, @S,
S,thesetofisolatedpointsofS, andthe
exteriorofS.
(a)
SD.1;2/[.2;3/[f4g[.7;1/
(b)
SDfallintegersg
(c)
SD[
˚
.n;nC1/
ˇ
ˇ
nDinteger
(d)
SD
˚
x
ˇ
ˇ
xD1=n;nD1;2;3;:::
12.
Prove:AlimitpointofasetSiseitheraninteriorpointoraboundarypointofS.
13.
Prove:AnisolatedpointofSisaboundarypointofS
c
.
14.
Prove:
(a)
AboundarypointofasetSiseitheralimitpointoranisolatedpointofS.
(b)
AsetSisclosedifandonlyifSD
S.
15.
Proveordisprove: Asethasnolimitpointsifandonlyifeachofits s pointsis
isolated.
16. (a)
Prove:IfSisboundedaboveandˇDsupS,thenˇ2@S.
(b)
Statetheanalogousresultforasetboundedbelow.
17.
Prove:IfSisclosedandbounded,theninfSandsupSarebothinS.
18.
IfanonemptysubsetSofRisbothopenandclosed,thenSDR.
19.
LetSbeanarbitraryset.Prove:
(a)
@Sisclosed.
(b)
S
0
isopen.
(c)
Theexterior
ofSisopen.
(d)
ThelimitpointsofSformaclosedset.
(e)
S
D
S.
20.
Givecounterexamplestothefollowingfalsestatements.
(a)
Theisolatedpointsofasetformaclosedset.
(b)
Everyopensetcontainsatleasttwopoints.
(c)
IfS
1
andS
2
arearbitrarysets,then@.S
1
[S
2
/D@S
1
[@S
2
.
(d)
IfS
1
andS
2
arearbitrarysets,then@.S
1
\S
2
/D@S
1
\@S
2
.
(e)
Thesupremumofaboundednonemptysetisthegreatestofitslimitpoints.
(f)
IfSisanyset,then@.@S/D@S.
(g)
IfSisanyset,then@
SD@S.
(h)
IfS
1
andS
2
arearbitrarysets,then.S
1
[S
2
/
0
DS
0
1
[S
0
2
.
Section1.3
TheRealLine
29
21.
LetSbeanonemptysubsetofRsuchthatifHisanyopencoveringofS,thenS
hasanopencovering
e
HcomprisedofﬁnitelymanyopensetsfromH. Showthat
Siscompact.
22.
AsetSis.inasetT ifST 
S.
(a)
Prove:IfSandT aresetsofrealnumbersandS T,thenSisdenseinT
ifandonlyifeveryneighborhoodofeachpointinT containsapointfromS.
(b)
Statehow
(a)
showsthatthedeﬁnitiongivenhereisconsistentwiththere-
23.
Prove:
(a)
.S
1
\S
2
/
0
DS
0
1
\S
0
2
(b)
S
0
1
[S
0
2
.S
1
[S
2
/
0
24.
Prove:
(a)
@.S
1
[S
2
/@S
1
[@S
2
(b)
@.S
1
\S
2
/@S
1
[@S
2
(c)
@
S@S
(d)
@SD@S
c
(e)
@.ST/@S[@T
CHAPTER2
DiﬀerentialCalculusof
FunctionsofOne Variable
INTHISCHAPTERwestudythedifferentialcalculusoffunctionsofonevariable.
SECTION2.1introducestheconceptoffunctionanddiscussesarithmeticoperationson
functions,limits,one-sidedlimits,limitsat˙1,andmonotonicfunctions.
SECTION2.2deﬁnescontinuityanddiscussesremovablediscontinuities,compositefunc-
tionalpropertiesofmonotonicfunctions.
SECTION2.3introducesthederivativeanditsgeometricinterpretation.Topicscoveredin-
cludetheinterchangeofdifferentiationandarithmeticoperations,thechainrule,one-sided
valuetheoremforderivatives,andthemeanvaluetheoremanditsconsequences.
SECTION2.4presentsacomprehensivediscussionofL’Hospital’srule.
SECTION2.5discussestheapproximationofafunctionf bytheTaylorpolynomialsof
f andappliesthisresulttolocatinglocalextremaoff. . Thesectionconcludeswiththe
extendedmeanvaluetheorem,whichimpliesTaylor’stheorem.
2.1FUNCTIONSANDLIMITS
Inthissectionwestudylimitsofreal-valuedfunctionsofarealvariable. Youstudied
limitsincalculus.However,wewilllookmorecarefullyatthedeﬁnitionoflimitandprove
theoremsusuallynotprovedincalculus.
Arulef thatassignstoeachmemberofanonemptysetDauniquememberofasetY
isafunctionfromDtoY. WewritetherelationshipbetweenamemberxofDandthe
memberyofYthatf assignstoxas
yDf.x/:
ThesetDisthedomainoff,denotedbyD
f
.ThemembersofY arethepossiblevalues
off.Ify
0
2Yandthereisanx
0
inDsuchthatf.x
0
/Dy
0
thenwesaythatf attains
30
Section2.1
FunctionsandLimits
31
orassumesthevaluey
0
.Thesetofvaluesattainedbyf istherangeoff.Areal-valued
functionofarealvariableisafunctionwhosedomainandrangearebothsubsetsofthe
reals.Althoughweareconcernedonlywithreal-valuedfunctionsofarealvariableinthis
section,ourdeﬁnitionsarenotrestrictedtothissituation.Inlatersectionswewillconsider
situationswheretherangeordomain,orboth,aresubsetsofvectorspaces.
Example2.1.1
Thefunctionsf,g,andhdeﬁnedon.1;1/by
f.x/Dx
2
; g.x/Dsinx; and h.x/De
x
haverangesŒ0;1/,Œ1;1,and.0;1/,respectively.
Example2.1.2
Theequation
Œf.x/
2
Dx
(2.1.1)
doesnotdeﬁneafunctionexceptonthesingletonsetf0g.Ifx<0,norealnumbersatisﬁes
Œf.x/
2
Dx and
f.x/0
deﬁneafunctionf onD
f
DŒ0;1/withvaluesf.x/D
p
x.Similarly,theconditions
Œg.x/
2
Dx and
g.x/0
deﬁneafunctiongonD
g
DŒ0;1/withvaluesg.x/D
p
x.Therangesoff andgare
Œ0;1/and.1;0,respectively.
Itisimportanttounderstandthatthedeﬁnitionofafunctionincludesthespeciﬁcation
suchas“thefunctionf withdomain.1;1/andvaluesf.x/Dx.”Wewillavoidthis
intwoways:(1)byagreeingthatifafunctionf isintroducedwithoutexplicitlydeﬁning
D
f
, thenD
f
willbeunderstoodtoconsistofallpointsx forwhichtheruledeﬁning
f.x/makessense,and(2)bybearinginmindthedistinctionbetweenf andf.x/,butnot
emphasizingitwhenitwouldbeanuisancetodoso.Forexample,wewillwrite“consider
thefunctionf.x/D
p
1x2,”ratherthan“considerthefunctionf deﬁnedonŒ1;1
byf.x/D
p
1x2,”or“considerthefunctiong.x/ D1=sinx,”ratherthan“consider
thefunctiongdeﬁnedforx¤k(kDinteger)byg.x/D1=sinx.”Wewillalsowrite
f Dc(constant)todenotethefunctionf deﬁnedbyf.x/Dcforallx.