Continuity and Uniform Continuity

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1、ContinuityandUniformContinuity521May12,20101.ThroughoutSwilldenoteasubsetoftherealnumbersRandf:S!RwillbearealvaluedfunctiondenedonS.ThesetSmaybeboundedlikeS=(0;5)=fx2R:0

2、enedbyaformula(orformulas).Denition2.ThefunctionfissaidtobecontinuousonSi8x02S8">09>08x2Sjxx0j<=)jf(x)f(x0)j<":Hencefisnotcontinuous1onSi9x02S9">08>09x2Sjxx0j<andjf(x)f(x0)j":Denition3.ThefunctionfissaidtobeuniformlycontinuousonSi8">09>08x02S8x2Sjx

3、x0j<=)jf(x)f(x0)j<":HencefisnotuniformlycontinuousonSi9">08>09x02S9x2Sjxx0j<andjf(x)f(x0)j":1ForanexampleofafunctionwhichisnotcontinuousseeExample22below.14.Theonlydierencebetweenthetwodenitionsistheorderofthequan-tiers.Whenyouprovefiscontinuousyourproofwi

4、llhavetheformChoosex02S.Choose">0.Let=(x0;").Choosex2S.Assumejxx0j<.


Thereforejf(x)f(x0)j<".Theexpressionfor(x0;")caninvolvebothx0and"butmustbeindependentofx.Theorderofthequaniersinthedenitionsignalsthis;intheproofxhasnotyetbeenchosenatthepointwhereisdenedso

5、thedenitionofmustnotinvolvex.(The


representtheproofthatjf(x)f(x0)j<"followsfromtheearlierstepsintheproof.)WhenyouprovefisuniformlycontinuousyourproofwillhavetheformChoose">0.Let=(").Choosex02S.Choosex2S.Assumejxx0j<.


Thereforejf(x)f(x0)j<".sotheexpressionfor

6、canonlyinvolve"andmustnotinvolveeitherxorx0.Itisobviousthatauniformlycontinuousfunctioniscontinuous:ifwecanndawhichworksforallx0,wecanndone(thesameone)whichworksforanyparticularx0.Wewillseebelowthattherearecontinuousfunctionswhicharenotuniformlycontinuous.Example5.L

7、etS=Randf(x)=3x+7.ThenfisuniformlycontinuousonS.Proof.Choose">0.Let="=3.Choosex02R.Choosex2R.Assumejxx0j<.Thenjf(x)f(x0)j=j(3x+7)(3x0+7)j=3jxx0j<3=":Example6.LetS=fx2R:00.Let="=8.Choosex02S.Choosex2S.Th

8、us0