133
Section6.3
TheInverseFunctionTheorem
409
Nowsupposethat>0.From(6.3.29),thereisaı
1
>0suchthat
jH
2
.X/j< if 0<jXX
0
jDjXG.U
0
/j<ı
1
:
(6.3.30)
SinceGiscontinuousatU
0
,thereisaı
2
2.0;ı/suchthat
jG.U/G.U
0
/j<ı
1
if 0<jUU
0
j<ı
2
:
Thisand(6.3.30)implythat
jH
1
.U/jDjH
2
.G.U//j< if 0<jUU
0
j<ı
2
:
Sincethisimplies(6.3.28),GisdifferentiableatX
0
.
SinceU
0
isanarbitrarymemberofF.N/, wecannowdropthezerosubscriptand
concludethatGiscontinuousanddifferentiableonF.N/,and
G
0
.U/DŒF
0
.X/
1
; U2F.N/:
ToseethatGiscontinuouslydifferentiableonF.N/,weobservethatbyTheorem6.1.14,
eachentryofG
0
.U/(thatis,eachpartialderivative@g
i
.U/=@u
j
,1 i;j n)canbe
writtenastheratio,withnonzerodenominator,ofdeterminantswithentriesoftheform
@f
r
.G.U//
@x
s
:
(6.3.31)
Since@f
r
=@x
s
iscontinuousonN andGiscontinuousonF.N/,Theorem5.2.10implies
that(6.3.31)iscontinuousonF.N/. Sinceadeterminantisacontinuousfunctionofits
entries,itnowfollowsthattheentriesofG
0
.U/arecontinuousonF.N/.
Branches oftheInverse
IfFisregularonanopensetS,wesaythatF1
S
isabranchof F1. . (Thisisaconvenient
terminologybutisnotmeanttoimplythatFactuallyhasaninverse.)Fromthisdefinition,
itispossibletodefineabranchofF
1
onasetT R.F/ifandonlyifTDF.S/,where
FisregularonS. TheremaybeopensubsetsofR.F/thatdonothavethisproperty,and
thereforenobranchofF
1
canbedefinedonthem. ItisalsopossiblethatT DF.S
1
/D
F.S
2
/,whereS
1
andS
2
aredistinctsubsetsofD
F
.Inthiscase,morethanonebranchof
F
1
isdefinedonT. Thus,wesawinExample6.3.4thattwobranchesofF
1
maybe
definedonasetT. InExample6.3.5infinitelymanybranchesofF
1
aredefinedonthe
sameset.
ItisusefultodefinebranchesoftheargumentTodothis,wethinkoftherelationship
betweenpolarandrectangularcoordinatesintermsofthetransformation
x
y
DF.r;/D
rcos
rsin
;
(6.3.32)
whereforthemomentweregardrandasrectangularcoordinatesofapointinanr-
plane.LetSbeanopensubsetoftherighthalfofthisplane(thatis,S
˚
.r;/
ˇ
ˇ
r>0
)