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Applications
43
102050Remark0
Itwouldbeperhapsmorecorrecttousethetermequi-
uniformlycontinuous,or,assomeauthorsdo,uniformlyequicontinuous.
Itis,however,moreconvenienttousetheshorterterm.
Noticethattheleapfromuniformcontinuitytoequicontinuityis
inasensesimilartotheonefromcontinuitytouniformcontinuity.
Inthecaseofflvanilla”continuity,weassumethatforeachpositive
8
andeverypointinthedomain,thereexistssome
δ>
0suchthatthe
well-knownconditionissatisfied.Inthecaseofuniformcontinuity,we
requirethatthesame
δ
worksforeverypointinthedomain.Inthecase
ofequicontinuity,weadditionallyconsiderawholefamilyoffunctions
andrequirethatthesame
δ
worksforeveryofthosefunctions(and
everypointoftheircommondomain).
Noticealsothatwhiletechnicallywecoulddefinepointwiseequicon-
tinuityofafamilyoffunctions,where
δ
maybedifferentforeachpoint
ofthedomain,butateachpointmuststaythesameforallfunctions
involved,wewillnotneedsuchanotion,andhencewearegoingto
usetheshorterterm,silentlyacknowledgingthatconveniencetrumps
logicalcorrectness.
102060Example0
Letusconsidertheset
V⊆C[
0,1
]
consistingoffunc-
tions
xn:[
0,1
]→R
definedbytheformulae
xn(t)=tn
for
n∈N
.
Notethatforany
δ∈(
0,1
)
wehave
|xn
0(
1
)lxn
0(
1
lδ)|=
1
l(
1
lδ)n0>
1
2,providedthatn0∈Nischosenlargeenoughsothat(1lδ)n0<1
2.
ThisclearlyshowsthatthesetVisnotequicontinuous.
102070Remark0
Thereasonwhytheset
V
fromtheexampleaboveisnot
equicontinuousissimple:itcontainsfunctionswitharbitrarilylarge
derivatives(indeed,
x!
n(
1
)=n
).Ontheotherhand,ifaset
W⊆C[
0,1
]
consistsofdifferentiablefunctionsandthereexistsaconstant
M>
0
suchthat
sup
|x!(t)|<M
forevery
x∈W
,then
W
isequicontinuous;
t∈[0,1]
toseethis,itsufficestoapplythemeanvaluetheorem.
Letusgeneralizethisabit.Supposethat
E
isaBanachspace,
W⊆C([a
,
b]
;
E)
andthereexistssomeconstant
M>
0suchthat
"x(t)lx(S)"<M|tlS|
forall
t
,
S∈[a
,
b]
and
x∈W
.(Mappings
satisfyingsuchaconditionarecalledLipschitzcontinuousandwillbe
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