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302 Chapter5
Real-ValuedFunctionsofSeveralVariables
27.
LetD
1
andD
2
becompactsubsetsofR
n
.Showthat
DD
˚
.X;Y/
ˇ
ˇ
X2D
1
;Y2D
2
isacompactsubsetofR
2n
.
28.
Prove:IfSisopenandSDA[Bwhere
A\BDA\
BD;,thenAandBare
open.
29.
GiveanexampleofaconnectedsetinR
n
thatisnotpolygonallyconnected.
30.
Provethataregionisconnected.
31.
ShowthattheintervalsaretheonlyregionsinR.
32.
Prove:AboundedsequenceinRhasaconvergentsubsequence.H
INT
:UseTheo-
rems5.1.14;4.2.2;and4.2.5.a/:
33.
Deﬁne“lim
r!1
X
r
D1”iffX
r
gisasequenceinR
n
,n2.
5.2 CONTINUOUS REAL-VALUEDFUNCTIONS OF
n
VARI-
ABLES
Wenowstudyreal-valuedfunctionsofnvariables.Wedenotethedomainofafunctionf
byD
f
andthevalueoff atapointXD.x
1
;x
2
;:::;x
n
/byf.X/orf.x
1
;x
2
;:::;x
n
/.
tionisdeﬁnedbyaformulasuchas
f.X/D
1x
2
1
x
2
2
x
2
n
1=2
(5.2.1)
or
g.X/D
1x
2
1
x
2
2
x
2
n
1
(5.2.2)
withoutspeciﬁcationofitsdomain, itistobeunderstoodthatitsdomainisthelargest
subsetofR
n
anyotherstipulation,thedomainoff in(5.2.1)istheclosedn-ball
˚
X
ˇ
ˇ
jXj1
,while
thedomainofgin(5.2.2)istheset
˚
X
ˇ
ˇ
jXj¤1
.
Themainobjectiveofthissectionistostudylimitsandcontinuityoffunctionsofn
variables.Theproofsofmanyofthetheoremsherearesimilartotheproofsoftheircoun-
terpartsinSections2.1and.Weleavemostofthemtoyou.
Deﬁnition5.2.1
Wesaythatf.X/approachesthelimitLasXapproachesX
0
and
write
lim
X!X
0
f.X/DL
ifX
0
isalimitpointofD
f
and,forevery>0,thereisaı>0suchthat
jf.X/Lj<
forallXinD
f
suchthat
0<jXX
0
j<ı:
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Section5.2
ContinuousReal-ValuedFunctionsof
n
Variables
303
Example5.2.1
If
g.x;y/D1x
2
2y
2
;
then
lim
.x;y/!.x
0
;y
0
/
g.x;y/D1x
2
0
2y
2
0
(5.2.3)
forevery.x
0
;y
0
/.Toseethis,wewrite
jg.x;y/.1x
2
0
2y
2
0
/jDj.1x
2
2y
2
/.1x
2
0
2y
2
0
/j
jx
2
x
2
0
jC2jy
2
y
2
0
j
Dj.xCx
0
/.xx
0
/jC2j.yCy
0
/.yy
0
/j
jXX
0
j.jxCx
0
jC2jyCy
0
/j/;
(5.2.4)
since
jxx
0
jjXX
0
j and jyy
0
jjXX
0
j:
IfjXX
0
j<1,thenjxj<jx
0
jC1andjyj<jy
0
jC1.Thisand(5.2.4)implythat
jg.x;y/.1x
2
0
2y
2
0
/j<KjXX
0
j if jXX
0
j<1;
where
KD.2jx
0
jC1/C2.2jy
0
jC1/:
Therefore,if>0and
jXX
0
j<ıDminf1;=Kg;
then
ˇ
ˇ
g.x;y/.1x
2
0
2y
2
0
/
ˇ
ˇ
<:
Thisproves(5.2.3).
Deﬁnition5.2.1doesnotrequirethatf bedeﬁnedatX
0
hoodofX
0
.
Example5.2.2
Thefunction
h.x;y/D
sin
p
1x22y2
p
1x22y2
isdeﬁnedonlyontheinterioroftheregionboundedbytheellipse
x
2
C2y
2
D1
(Figure5.2.1
(a)
,page304). Itisnotdeﬁnedatanypointoftheellipseitselforonany
deletedneighborhoodofsuchapoint.Nevertheless,
lim
.x;y/!.x
0
;y
0
/
h.x;y/D1
(5.2.5)
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304 Chapter5
Real-ValuedFunctionsofSeveralVariables
if
x
2
0
C2y
2
0
D1:
(5.2.6)
Toseethis,let
u.x;y/D
p
1x22y2:
Then
h.x;y/D
sinu.x;y/
u.x;y/
:
(5.2.7)
Recallthat
lim
r!0
sinr
r
D1I
therefore,if>0,thereisaı
1
>0suchthat
ˇ
ˇ
ˇ
ˇ
sinu
u
1
ˇ
ˇ
ˇ
ˇ
< if 0<juj<ı
1
:
(5.2.8)
From(5.2.3),
lim
.x;y/!.x
0
;y
0
/
.1x
2
2y
2
/D0
if(5.2.6)holds,sothereisaı>0suchthat
0<u
2
.x;y/D.1x
2
2y
2
/<ı
2
1
ifXD.x;y/isintheinterioroftheellipseandjXX
0
regionofFigure5.2.1
(b)
.
Therefore,
0<uD
p
1x22y
1
(5.2.9)
ifXisintheinterioroftheellipseandjXX
0
Figure5.2.1
(b)
.This,(5.2.7),and(5.2.8)implythat
jh.x;y/1j<
forsuchX,whichimplies(5.2.5).
(a)
y
x
x2+2y=1
(b)
y
x
x2+2y=1
X−X
0
=  δ
X
0
Figure5.2.1
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Section5.2
ContinuousReal-ValuedFunctionsof
n
Variables
305
ThefollowingtheoremisanalogoustoTheorem2.1.3.Weleaveitsprooftoyou(Exer-
cise5.2.2).
Theorem5.2.2
Iflim
X!X
0
f.X/exists;thenitisunique.
WheninvestigatingwhetherafunctionhasalimitatapointX
0
,norestrictioncanbe
0
, exceptthatXmustbeinD
f
. Thenext
Example5.2.3
Thefunction
f.x;y/D
xy
x2Cy2
isdeﬁnedeverywhereinR
2
exceptat.0;0/.Doeslim
.x;y/!.0;0/
f.x;y/exist?Ifwetry
functionalvalues
f.x;x/D
x
2
2x2
D
1
2
andconcludethatthelimitis1=2.However,ifwelet.x;y/approach.0;0/alongtheline
yDx,weseethefunctionalvalues
f.x;x/D
x
2
2x2
D
1
2
andconcludethatthelimitequals 1=2. From m Theorem 5.2.2, thesetwoconclusions
cannotbothbecorrect.Infact,theyarebothincorrect.Whatwehaveshownisthat
lim
x!0
f.x;x/D
1
2
and
lim
x!0
f.x;x/D
1
2
:
Sincelim
x!0
f.x;x/andlim
x!0
f.x;x/mustbothequallim
.x;y/!.0;0/
f.x;y/ifthe
latterexists(Exercise5.2.3
(a)
),weconcludethatthelatterdoesnotexist.
Thesum,difference, andproductoffunctionsofnvariablesaredeﬁnedinthesame
wayastheyareforfunctionsofonevariable(Deﬁnition2.1.1),andtheproofofthenext
theoremisthesameastheproofofTheorem2.1.4.
Theorem5.2.3
Supposethatf andgaredeﬁnedonasetD;X
0
isalimitpointof
D;and
lim
X!X
0
f.X/DL
1
;
lim
X!X
0
g.X/DL
2
:
Then
lim
X!X
0
.f Cg/.X/DL
1
CL
2
;
(5.2.10)
lim
X!X
0
.fg/.X/DL
1
L
2
;
(5.2.11)
lim
X!X
0
.fg/.X/DL
1
L
2
;
(5.2.12)
and;ifL
2
¤0;
lim
X!X
0
f
g
.X/D
L
1
L
2
:
(5.2.13)
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306 Chapter5
Real-ValuedFunctionsofSeveralVariables
Inﬁnite LimitsandLimitsas
jXj!1
Deﬁnition5.2.4
Wesaythatf.X/approaches1asXapproachesX
0
andwrite
lim
X!X
0
f.X/D1
ifX
0
isalimitpointofD
f
and,foreveryrealnumberM,thereisaı>0suchthat
f.X/>M
whenever 0<jXX
0
j<ı and X2D
f
:
Wesaythat
lim
X!X
0
f.X/D1
if
lim
X!X
0
.f/.X/D1:
Example5.2.4
If
f.X/D.1x
2
1
x
2
2
x
2
n
/
1=2
;
then
lim
X!X
0
f.X/D1
ifjX
0
jD1,because
f.X/D
1
jXX
0
j
;
so
f.X/>M
if 0<jXX
0
j<ıD
1
M
:
Example5.2.5
If
f.x;y/D
1
xC2yC1
;
thenlim
.x;y/!.1;1/
f.x;y/doesnotexist(whynot?),but
lim
.x;y/!.1;1/
jf.x;y/jD1:
Toseethis,weobservethat
jxC2yC1jDj.x1/C2.yC1/j
p
5jXX
0
j (bySchwarz’sinequality),
whereX
0
D.1;1/,so
jf.x;y/jD
1
jxC2yC1j
1
p
5jXX
0
j
:
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Section5.2
ContinuousReal-ValuedFunctionsof
n
Variables
307
Therefore,
jf.x;y/j>M
if 0<jXX
0
j<
1
M
p
5
:
Example5.2.6
Thefunction
f.x;y;´/D
ˇ
ˇ
ˇ
ˇ
sin
1
x2Cy2
ˇ
ˇ
ˇ
ˇ
x2Cy22
assumesarbitrarilylargevaluesineveryneighborhoodof.0;0;0/.Forexample,ifX
k
D
.x
k
;y
k
k
/,where
x
k
Dy
k
k
D
1
q
3
kC
1
2
;
then
f.X
k
/D
kC
1
2
:
However,thisdoesnotimplythatlim
X!0
f.X/D1,since,forexample,everyneighbor-
hoodof.0;0;0/alsocontainspoints
X
k
D
1
p
3k
;
1
p
3k
;
1
p
3k
forwhichf.
X
k
/D0.
Deﬁnition5.2.5
IfD
f
isunbounded;wesaythat
lim
jXj!1
f.X/DL (ﬁnite)
ifforevery>0,thereisanumberRsuchthat
jf.X/Lj< whenever jXjR
and X2D
f
:
Example5.2.7
If
f.x;y;´/Dcos
1
x2C2y22
;
then
lim
jXj!1
f.X/D1:
(5.2.14)
Toseethis,werecallthatthecontinuityofcosuatuD0impliesthatforeach>0there
isaı>0suchthat
jcosu1j< if juj<ı:
308 Chapter5
Real-ValuedFunctionsofSeveralVariables
Since
1
x2C2y22
1
jXj2
;
itfollowsthatifjXj>1=
p
ı,then
1
x2C2y22
<ı:
Therefore,
jf.X/1j<:
Thisproves(5.2.14).
Example5.2.8
Considerthefunctiondeﬁnedonlyonthedomain
DD
˚
.x;y/
ˇ
ˇ
0<yax
; 0<a<1
(Figure5.2.2),by
f.x;y/D
1
xy
:
Wewillshowthat
lim
jXj!1
f.x;y/D0:
(5.2.15)
Itisimportanttokeepinmindthatweneedonlyconsider.x;y/inD, sincef isnot
deﬁnedelsewhere.
InD,
xyx.1a/
(5.2.16)
and
jXj
2
Dx
2
Cy
2
x
2
.1Ca
2
/;
so
x
jXj
p
1Ca2
:
Thisand(5.2.16)implythat
xy
1a
p
1Ca2
jXj; X2D;
so
jf.x;y/j
p
1Ca2
1a
1
jXj
; X2D:
Therefore,
jf.x;y/j<
ifX2Dand
jXj>
p
1Ca2
1a
1
:
Thisimplies(5.2.15).
Section5.2
ContinuousReal-ValuedFunctionsof
n
Variables
309
y
x
y = ax
Figure5.2.2
Weleaveittoyoutodeﬁnelim
jXj!1
f.X/ D1andlim
jXj!1
f.X/ D1(Exer-
cise5.2.6).
X!X
0
f.X/exists”means
thatlim
X!X
0
f.X/DL,whereLisﬁnite;toleaveopenthepossibilitythatLD˙1,we
willsaythat“lim
X!X
0
f.X/existsintheextendedreals.”Asimilarconventionappliesto
limitsasjXj!1.
Theorem5.2.3remainsvalidif“lim
X!X
0
”isreplacedby“lim
jXj!1
,” providedthat
Disunbounded. Moreover,(5.2.10),(5.2.11),and(5.2.12)arevalidineitherversionof
Theorem5.2.3ifeitherorbothofL
1
andL
2
isinﬁnite,providedthattheirrightsidesare
notindeterminate,and(5.2.13)remainsvalidifL
2
¤0andL
1
=L
2
isnotindeterminate.
Continuity
Wenowdeﬁnecontinuityforfunctionsofnvariables.Thedeﬁnitionisquitesimilartothe
deﬁnitionforfunctionsofonevariable.
Deﬁnition5.2.6
IfX
0
isinD
f
andisalimitpointofD
f
,thenwesaythatf is
continuousatX
0
if
lim
X!X
0
f.X/Df.X
0
/:
ThenexttheoremfollowsfromthisandDeﬁnition5.2.1.
310 Chapter5
Real-ValuedFunctionsofSeveralVariables
Theorem5.2.7
SupposethatX
0
isinD
f
andisalimitpointofD
f
:Thenf iscon-
tinuousatX
0
ifandonlyifforeach>0thereisaı>0suchthat
jf.X/f.X
0
/j<
whenever
jXX
0
j<ı and X2D
f
:
InapplyingthistheoremwhenX
0
2D
0
f
,wewillusuallyomit“andX2D
f
,”itbeing
understoodthatS
ı
.X
0
/D
f
.
Wewillsaythatf iscontinuousonSiff iscontinuousateverypointofS.
Example5.2.9
FromExample5.2.1,wenowseethatthefunction
f.x;y/D1x
2
2y
2
iscontinuousonR
2
.
Example5.2.10
IfweextendthedeﬁnitionofhinExample5.2.2sothat
h.x;y/D
8
ˆ
<
ˆ
:
sin
p
1x22y2
p
1x22y2
; x
2
C2y
2
<1;
1;
x
2
C2y
2
D1;
thenitfollowsfromExample5.2.2thathiscontinuousontheellipse
x
2
C2y
2
D1:
WewillseeinExample5.2.13thathisalsocontinuousontheinterioroftheellipse.
Example5.2.11
Itisimpossibletodeﬁnethefunction
f.x;y/D
xy
x2Cy2
attheorigintomakeitcontinuousthere,sincewesawinExample5.2.3that
lim
.x;y/!.0;0/
f.x;y/
doesnotexist.
Theorem5.2.3impliesthenexttheorem,whichisanalogoustoTheorem2.2.5and,like
thelatter,permitsustoinvestigatecontinuityofagivenfunctionbyregardingthefunction