微积分讲义chap31

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1、Chapter3ContinuityonR3.1Twosidelimits3.1Definition.LetIR,IbeanopenintervalandaI.SupposethatfisarealfunctiondefinedeverywhereonIexceptpossiblyata.Wesaythatf(x)convergestoLasxapproachestoaifgiven>0,>0s.tfor0<.Remarks.1.fmaybedefinedonIexpectpossiblya.2.Iff(a)isdefinedthenf(a

2、)&Lmaybeindependent.3.Wedenoteitby=L3.2Example.Supposef(x)=mx+bforsomem,bR,provef(a)=forallaR.Proof.Given>0,find>0s.tfor0<.Assume

3、).For0<<<4<<.3.4Remark.Supposethatf&garereal-valuedfunctionsdefinedonopenintervalIofRexceptpossiblyaI.Iff=gonI{a},then=providedthelimitsexist.Proof:Suppose=LGiven>0,>0s.tfor0

4、xample3.2.3.6.Theorem(SequentialCharacterizationoflimits)I:openintervalofR,aI.fisdefinedonIexceptpossiblya.Then=Liff=Lforanysequences{}whichconvergestoaProof.“”Let{}beasequenceofRwithlimita.Weshowthat=L.given>0,findanNNs.t0s.t0

5、“”SupposenotL.i.e.>0and>0,thereisanxs.t0

6、pposethatf&garereal-valuedfunctionsdefinedonopenintervalIofRexceptpossiblyaI.Ifandexist,then=+Proof:LetL=andM=ByTheorem3.6,ifI{a}convergestoa,thenf()Landg()Masn.ByTheorem2.12(f+g)()=f()+g()L+MasnByTheorem3.6,=L+M=+3.9Theorem[SqueezeTheoremforfunction]Supposethatf,gandhare

7、real-valuefunctionsdefinedonanopenintervalIexceptpossiblyaI.Iff(x)g(x)h(x)forxI{a}and==L,then=L.Proof.=Lgiven>0,>0s.tfor00,>0s.tfor0

8、ereal-valuefunctiondefinedonanopenintervalIexceptpossiblyaI.Iff(x)g(x),xI{a},andexist.,T