pdfsharp c# : Add hyperlink in pdf software application cloud windows html wpf class TRENCH_REAL_ANALYSIS6-part279

Section2.2
Continuity
53
(a)
Showthat
lim
x!x
0
.f Cg/.x/
lim
x!x
0
f.x/C
lim
x!x
0
g.x/:
(b)
Showthat
lim
x!x
0
.f Cg/.x/ lim
x!x
0
f.x/C lim
x!x
0
g.x/:
(c)
Stateinequalitiesanalogoustothosein
(a)
and
(b)
for
lim
x!x
0
.f g/.x/ and
lim
x!x
0
.f g/.x/:
38.
Prove:lim
x!x
0
f.x/exists(finite)ifandonlyifforeach>0thereisaı>0
suchthatjf.x
1
/f.x
2
/j< ifx
0
ı <x
1
,x
2
<x
0
. H
INT
:Forsufficiency;
showthatf isboundedonsomeinterval.a;x
0
/and
lim
x!0
f.x/D
lim
x!x
0
f.x/:
ThenuseExercise2.1.36.c/:
39.
Supposethatfisboundedonaninterval.x
0
;b.UsingDefinition2.1.10asaguide,
define
lim
x!x
0
C
f.x/(therightlimitsuperioroffatx
0
)andlim
x!x
0
C
f.x/(the
rightlimitinferioroff atx
0
).Thenprovethattheyexist.H
INT
:UseTheorem2.1.9:
40.
Supposethatf isboundedonaninterval.x
0
;b. Showthatlim
x!x
0
C
f.x/ D
lim
x!x
0
C
f.x/ifandonlyiflim
x!x
0
C
f.x/exists,inwhichcase
lim
x!x
0
C
f.x/D
lim
x!x
0
C
f.x/D
lim
x!x
0
C
f.x/:
41.
Supposethatfisboundedonanopenintervalcontainingx
0
.Showthatlim
x!x
0
f.x/
existsifandonlyif
lim
x!x
0
f.x/D
lim
x!x
0
C
f.x/D
lim
x!x
0
f.x/D
lim
x!x
0
C
f.x/;
inwhichcaselim
x!x
0
f.x/isthecommonvalueofthesefourexpressions.
2.2CONTINUITY
Inthissectionwestudycontinuousfunctionsofarealvariable.Wewillprovesomeimpor-
tanttheoremsaboutcontinuousfunctionsthat,althoughintuitivelyplausible,arebeyond
thescopeoftheelementarycalculuscourse.Theyareaccessiblenowbecauseofourbetter
understandingoftherealnumbersystem,especiallyofthosepropertiesthatstemfromthe
completenessaxiom.
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54 Chapter2
DifferentialCalculusofFunctionsofOneVariable
Thedefinitionsof
f.x
0
/D
lim
x!x
0
f.x/; f.x
0
C/D
lim
x!x
0
C
f.x/; and
lim
x!x
0
f.x/
donotinvolvef.x
0
/orevenrequirethatitbedefined.However,thecasewheref.x
0
/is
definedandequaltooneormoreofthesequantitiesisimportant.
Definition2.2.1
(a)
Wesaythatfiscontinuousatx
0
iffisdefinedonanopeninterval.a;b/containing
x
0
andlim
x!x
0
f.x/Df.x
0
/.
(b)
Wesaythatf iscontinuousfromtheleftatx
0
iff isdefinedonanopeninterval
.a;x
0
/andf.x
0
/Df.x
0
/.
(c)
Wesaythatf iscontinuousfromtherightatx
0
iff isdefinedonanopeninterval
.x
0
;b/andf.x
0
C/Df.x
0
/.
Thefollowingtheoremprovidesamethodfordeterminingwhetherthesedefinitionsare
satisfied. Theproof,whichweleavetoyou(Exercise2.2.1),restsonDefinitions2.1.2,
2.1.5,and2.2.1.
Theorem2.2.2
(a)
Afunctionfiscontinuousatx
0
ifandonlyiff isdefinedonanopeninterval.a;b/
containingx
0
andforeach>0thereisaı>0suchthat
jf.x/f.x
0
/j<
(2.2.1)
wheneverjxx
0
j<ı:
(b)
Afunctionf iscontinuousfromtherightatx
0
ifandonlyiff isdefinedonan
intervalŒx
0
;b/andforeach>0thereisaı>0suchthat(2.2.1)holdswhenever
x
0
x<x
0
Cı:
(c)
Afunctionfiscontinuousfromtheleftatx
0
ifandonlyiffisdefinedonaninterval
.a;x
0
andforeach>0
thereisaı>0suchthat(2.2.1)holdswheneverx
0
ı<xx
0
:
FromDefinition2.2.1andTheorem2.2.2,f is
continuousatx
0
ifandonlyif
f.x
0
/Df.x
0
C/Df.x
0
/
or,equivalently,ifandonlyifitiscontinuousfromtherightandleftatx
0
(Exercise2.2.2).
Example2.2.1
Letf bedefinedonŒ0;2by
f.x/D
x
2
;
0x<1;
xC1; 1x2
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Section2.2
Continuity
55
(Figure2.2.1);then
f.0C/D0Df.0/;
f.1/D1¤f.1/D2;
f.1C/D2Df.1/;
f.2/D3Df.2/:
Therefore,f iscontinuousfromtherightat0and1andcontinuousfromtheleftat2,but
notat1.If0<x,x
0
<1,then
jf.x/f.x
0
/jDjx
2
x
2
0
jDjxx
0
jjxCx
0
j
2jxx
0
j< if
jxx
0
j<=2:
Hence,fiscontinuousateachx
0
in.0;1/.If1<x,x
0
<2,then
jf.x/f.x
0
/jDj.xC1/.x
0
C1/Djxx
0
j
< if
jxx
0
j<:
Hence,fiscontinousateachx
0
in.1;2/.
2
3
2
1
1
y
x
y = x + 1,  1 ≤ x ≤ 2
y = x2,  0 ≤ x < 1
Figure2.2.1
Definition2.2.3
Afunctionf iscontinuousonanopeninterval.a;b/ifitiscontinu-
ousateverypointin.a;b/.If,inaddition,
f.b/Df.b/
(2.2.2)
or
f.aC/Df.a/
(2.2.3)
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56 Chapter2
DifferentialCalculusofFunctionsofOneVariable
thenf iscontinuouson.a;borŒa;b/, , respectively. Iff iscontinuouson.a;b/and
(2.2.2)and(2.2.3)bothhold,thenfiscontinuousonŒa;b.Moregenerally,ifSisasubset
ofD
f
consistingoffinitelyorinfinitelymanydisjointintervals,thenf iscontinuousonS
iff iscontinuousoneveryintervalinS.(Henceforth,inconnectionwithfunctionsofone
variable,wheneverwesay“f iscontinuousonS”wemeanthatSisasetofthiskind.)
Example2.2.2
Letf.x/D
p
x,0x<1.Then
jf.x/f.0/jD
p
x< if 0x<
2
;
sof.0C/Df.0/.Ifx
0
>0andx0,then
jf.x/f.x
0
/jDj
p
x
p
x
0
jD
jxx
0
j
p
xC
p
x
0
jxx
0
j
p
x
0
< if jxx
0
j<
p
x
0
;
solim
x!x
0
f.x/Df.x
0
/.Hence,f iscontinuousonŒ0;1/.
Example2.2.3
Thefunction
g.x/D
1
sinx
iscontinuousonS D
S
1
nD1
.n;nC1/. However,gisnotcontinuousatanyx
0
D n
(integer),sinceitisnotdefinedatsuchpoints.
Thefunctionf definedinExample2.2.1(seealsoFigure2.2.1)iscontinuousonŒ0;1/
andŒ1;2,butnotonanyopenintervalcontaining1.Thediscontinuityoff thereisofthe
simplestkind,describedinthefollowingdefinition.
Definition2.2.4
Afunctionf ispiecewisecontinuousonŒa;bif
(a)
f.x
0
C/existsforallx
0
inŒa;b/;
(b)
f.x
0
/existsforallx
0
in.a;b;
(c)
f.x
0
C/Df.x
0
/Df.x
0
/forallbutfinitelymanypointsx
0
in.a;b/.
If
(c)
failstoholdatsomex
0
in.a;b/,f hasajumpdiscontinuityatx
0
. Also,f hasa
jumpdiscontinuityataiff.aC/¤f.a/oratbiff.b/¤f.b/.
Example2.2.4
Thefunction
f.x/D
8
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
<
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
:
1; xD0;
x; 0<x<1;
2; xD1;
x; 1<x2;
1; 2<x<3;
0; xD3;
(Figure2.2.2)isthegraphofapiecewisecontinuousfunctiononŒ0;3,withjumpdiscon-
tinuitiesatx
0
D0,1,2,and3.
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Section2.2
Continuity
57
2
3
2
3
1
1
−1
y
x
Figure2.2.2
Thereasonfortheadjective“jump”canbeseeninFigures2.2.1and2.2.2,wherethe
graphsexhibitadefinitejumpateachpointofdiscontinuity.Thenextexampleshowsthat
notalldiscontinuitiesareofthiskind.
Example2.2.5
Thefunction
f.x/D
8
ˆ
<
ˆ
:
sin
1
x
; x¤0;
0;
xD0;
iscontinuousatallx
0
exceptx
0
D0.Asxapproaches0fromeitherside,f.x/oscillates
between1and1withever-increasingfrequency, soneitherf.0C/norf.0/ exists.
Therefore,thediscontinuityoff at0isnotajumpdiscontinuity,andif>0,thenf f is
notpiecewisecontinuousonanyintervaloftheformŒ;0,Œ;,orŒ0;.
Theorems2.1.4and2.2.2implythenexttheorem(Exercise2.2.18).
Theorem2.2.5
Iff andgarecontinuousonasetS;thensoarefCg;f g;and
fg:Inaddition;f=giscontinuousateachx
0
inSsuchthatg.x
0
/¤0:
Example2.2.6
Sincetheconstantfunctionsandthefunctionf.x/Dxarecontinu-
ousforallx,successiveapplicationsofthevariouspartsofTheorem2.2.5implythatthe
function
r.x/D
9x
2
xC1
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58 Chapter2
DifferentialCalculusofFunctionsofOneVariable
iscontinuousforallxexceptx D1(seeExample2.1.7). Moregenerally,bystarting
fromTheorem2.2.5andusing
induction,itcanbeshownthatiff
1
,f
2
,...,f
n
arecontinuousonasetS,thensoare
f
1
Cf
2
CCf
n
andf
1
f
2
f
n
.Therefore,anyrationalfunction
r.x/D
a
0
Ca
1
xCCa
n
xn
b
0
Cb
1
xCCb
m
xm
.b
m
¤0/
iscontinuousforallvaluesofxexceptthoseforwhichitsdenominatorvanishes.
RemovableDiscontinuities
Letf bedefinedonadeletedneighborhoodofx
0
anddiscontinuous(perhapsevenunde-
fined)atx
0
.Wesaythatf hasaatx
0
iflim
x!x
0
f.x/exists.Inthiscase,thefunction
g.x/D
8
<
:
f.x/
ifx2D
f
andx¤x
0
;
lim
x!x
0
f.x/ ifxDx
0
;
iscontinuousatx
0
.
Example2.2.7
Thefunction
f.x/Dxsin
1
x
isnotdefinedatx
0
D0,andthereforecertainlynotcontinuousthere,butlim
x!0
f.x/D0
(Example2.1.6).Therefore,f hasaremovablediscontinuityat0.
Thefunction
f
1
.x/Dsin
1
x
isundefinedat0anditsdiscontinuitythereisnotremovable,sincelim
x!0
f
1
.x/doesnot
exist(Example2.2.5).
Composite Functions
Wehaveseenthattheinvestigationoflimitsandcontinuitycanbesimplifiedbyregardinga
givenfunctionastheresultofaddition,subtraction,multiplication,anddivisionofsimpler
functions.Anotheroperationusefulinthisconnectioniscompositionoffunctions;thatis,
substitutionofonefunctionintoanother.
Definition2.2.6
Supposethatf andgarefunctionswithdomainsD
f
andD
g
. If
D
g
hasanonemptysubsetT suchthatg.x/2D
f
wheneverx 2T,thenthecomposite
functionfıgisdefinedonT by
.f ıg/.x/Df.g.x//:
Section2.2
Continuity
59
Example2.2.8
If
f.x/Dlogx and g.x/D
1
1x2
;
then
D
f
D.0;1/ and
D
g
D
˚
x
ˇ
ˇ
x¤˙1
:
Sinceg.x/>0ifx2T D.1;1/,thecompositefunctionfıgisdefinedon.1;1/by
.f ıg/.x/Dlog
1
1x2
:
Weleaveittoyoutoverifythatgıf isdefinedon.0;1=e/[.1=e;e/[.e;1/by
.gıf/.x/D
1
1.logx/2
:
Thenexttheoremsaysthatthecompositionofcontinuousfunctionsiscontinuous.
Theorem2.2.7
Supposethatgiscontinuousatx
0
;g.x
0
/isaninteriorpointofD
f
;
andf iscontinuousatg.x
0
/:Thenfıgiscontinuousatx
0
:
Proof
Supposethat>0.Sinceg.x
0
/isaninteriorpointofD
f
andf iscontinuous
atg.x
0
/,thereisaı
1
>0suchthatf.t/isdefinedand
jf.t/f.g.x
0
//j< if jtg.x
0
/j<ı
1
:
(2.2.4)
Sincegiscontinuousatx
0
,thereisaı>0suchthatg.x/isdefinedand
jg.x/g.x
0
/j<ı
1
if jxx
0
j<ı:
(2.2.5)
Now(2.2.4)and(2.2.5)implythat
jf.g.x//f.g.x
0
//j< if jxx
0
j<ı:
Therefore,f ıgiscontinuousatx
0
.
SeeExercise2.2.22forarelatedresultconcerninglimits.
Example2.2.9
InExamples2.2.2and2.2.6wesawthatthefunction
f.x/D
p
x
iscontinuousforx>0,andthefunction
g.x/D
9x
2
xC1
iscontinuousforx ¤1. Sinceg.x/ / > 0ifx < 3or1< x < 3,Theorem2.2.7
impliesthatthefunction
.f ıg/.x/D
s
9x2
xC1
iscontinuouson.1;3/[.1;3/.Itisalsocontinuousfromtheleftat3and3.
60 Chapter2
DifferentialCalculusofFunctionsofOneVariable
BoundedFunctions
Afunctionf isboundedbelowonasetSifthereisarealnumbermsuchthat
f.x/mforallx2S:
Inthiscase,theset
V D
˚
f.x/
ˇ
ˇ
x2S
hasaninfimum˛,andwewrite
˛D inf
x2S
f.x/:
Ifthereisapointx
1
inSsuchthatf.x
1
/D˛,wesaythat˛istheminimumoff onS,
andwrite
˛Dmin
x2S
f.x/:
Similarly,f isboundedaboveonSifthereisarealnumberMsuchthatf.x/M for
allxinS.Inthiscase,V hasasupremumˇ,andwewrite
ˇDsup
x2S
f.x/:
Ifthereisapointx
2
inSsuchthatf.x
2
/Dˇ,wesaythatˇisthemaximumoff onS,
andwrite
ˇDmax
x2S
f.x/:
Iff isboundedaboveandbelowonasetS,wesaythatf isboundedonS.
Figure2.2.3illustratesthegeometric meaningofthesedefinitionsfora functionf
boundedonanintervalS D D Œa;b. Thegraphoff lies s inthestripboundedbythe
linesy D D M andy D m, , whereM isany y upperboundandmisany lowerbound
forf onŒa;b. Thenarroweststripcontainingthegraphistheoneboundedaboveby
yDˇDsup
axb
f.x/andbelowbyyD˛Dinf
axb
f.x/.
y
x
y = α
y = β
y = m
y = M
Figure2.2.3
Section2.2
Continuity
61
Example2.2.10
Thefunction
g.x/D
(
1
2
;
xD0 or xD1;
1x; 0<x<1;
C
(Figure2.2.4
(a)
)isboundedonŒ0;1,and
sup
0x1
g.x/D1;
inf
0x1
g.x/D0:
Therefore,ghasnomaximumorminimumonŒ0;1,sinceitdoesnotassumeeitherofthe
values0and1.
Thefunction
h.x/D1x; 0x1;
whichdiffersfromgonlyat0and1(Figure2.2.4
(b)
),hasthesamesupremumandinfi-
mumasg,butitattainsthesevaluesatxD0andxD1,respectively;therefore,
max
0x1
h.x/D1 and
min
0x1
h.x/D0:
2
1
1
1
y
x
1
1
y
x
(a)
(b)
y = g(x)
y = 1 − x
Figure2.2.4
Example2.2.11
Thefunction
f.x/De
x.x1/
sin
1
x.x1/
; 0<x<1;
oscillatesbetween˙e
x.x1/
infinitelyoftenineveryintervaloftheform.0;/or.1;1/,
where0<<1,and
sup
0<x<1
f.x/D1;
inf
0<x<1
f.x/D1:
However,f doesnotassumethesevalues,sof hasnomaximumorminimumon.0;1/.
62 Chapter2
DifferentialCalculusofFunctionsofOneVariable
Theorem2.2.8
IffiscontinuousonafiniteclosedintervalŒa;b;thenfisbounded
onŒa;b:
Proof
Supposethatt 2Œa;b. Sincef iscontinuousatt,thereisanopenintervalI
t
containingtsuchthat
jf.x/f.t/j<1 if
x2I
t
\Œa;b:
(2.2.6)
(Toseethis,setD1in(2.2.1),Theorem2.2.2.)ThecollectionHD
˚
I
t
ˇ
ˇ
atb
is
anopencoveringofŒa;b. SinceŒa;biscompact,theHeine–Boreltheoremimpliesthat
therearefinitelymanypointst
1
,t
2
,..., t
n
suchthattheintervalsI
t
1
,I
t
2
,..., I
t
n
cover
Œa;b.Accordingto(2.2.6)withtDt
i
,
jf.x/f.t
i
/j<1 if
x2I
t
i
\Œa;b:
Therefore,
jf.x/jDj.f.x/f.t
i
//Cf.t
i
/jjf.x/f.t
i
/jCjf.t
i
/j
1Cjf.t
i
/j if
x2I
t
i
\Œa;b:
(2.2.7)
Let
MD1C max
1in
jf.t
i
/j:
SinceŒa;b
S
n
iD1
I
t
i
\Œa;b
,(2.2.7)impliesthatjf.x/jMifx2Œa;b.
ThisproofillustratestheutilityoftheHeine–Boreltheorem,whichallowsustochoose
Masthelargestofafinitesetofnumbers.
Theorem2.2.8andthecompletenessoftherealsimplythat
iff iscontinuousonafiniteclosedintervalŒa;b,thenf hasaninfimumandasupre-
mumonŒa;b. Thenexttheoremshowsthatf actuallyassumes s thesevaluesatsome
pointsinŒa;b.
Theorem2.2.9
Supposethatf iscontinuousonafiniteclosedintervalŒa;b:Let
˛D
inf
axb
f.x/ and ˇD sup
axb
f.x/:
Then˛andˇarerespectivelytheminimumandmaximumoff onŒa;bIthatis;thereare
pointsx
1
andx
2
inŒa;bsuchthat
f.x
1
/D˛ and f.x
2
/Dˇ:
Proof
Weshowthatx
1
existsandleaveittoyoutoshowthatx
2
exists(Exercise2.2.24).
Supposethatthereisnox
1
inŒa;bsuchthatf.x
1
/ D ˛. Thenf.x/ > > ˛forall
x2Œa;b.Wewillshowthatthisleadstoacontradiction.
Supposethatt2Œa;b.Thenf.t/>˛,so
f.t/>
f.t/C˛
2
>˛:
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