PDF-2IntegrationforcontinuousfunctionTheorem2.1.Letf:[a;b]!Rbecontinuouson

nx2a2ba nxnbgThenZbaflimn1UfPnlimn1LfPnProofItsucestoshowthatlimn1UfPnLfPn0sinceexercise295in1willthenimplytheresultLet0begivenSincefisuniformlycontinuouson. Asaconsequence,compositionofcontinuousmapsdenesafunction[X;Y][Y;Z]![X;Z];([f];[g])7![gf]:2.HomotopyequivalencesDenition2.1.Letf:X!Ybeacontinuousmap.Thenfissaidtobehomotopyequivalenceifthereexistsa 4Note:Apolynomialofdegree4mayhavenozeros,butstillfactor(asaproductofquadraticorhigher-degreepolynomials).Tocheckforquadraticorhigher-degreefactors,uselongdivisionwitheachofthepotentialfactors(hopeful Nowthesethreeconditionsareveriedtoshowthatagiventotalextensionxof isneitherr.e.traceablenorautocomplex.Assumearecursiveboundhbegiven.Letf(m)=C(x2h(m+1)+m+4).Choosem;nsuchthath(m)+m+3nh(m+1)+m+4and 4KEITHCONRADProof.Letf(T)2OK[T]beEisensteinatsomeprimeideal.Iff(T)isreducibleinK[T]thenf(T)=g(T)h(T)forsomenonconstantg(T)andh(T)inK[T].WerstshowthatgandhcanbechoseninOK[T].Asfismonic,wecanassumegand permutationsinSnisdenotedbyAn.CorollaryA.4.Thefollowinghold:(1)AnisasubgroupofSn.(2)Theindex[Sn:An]=2(assumingn2).ThisisequivalenttosayingthatexactlyhalfofallpermutationsinSnareeven.Proof.(1)Thisfoll ; x]andy=[y ; y],theinclusionisdenedbyxvy,y 6x ^ x6 y:Theinclusionisthenrelatedtothedualintervalbyxvy,dualxwdualy.Definition1.Letf:Rn!Rbeacontinuousfunctionandx2IKn,whichwecandecomposeinxA2IRpandxE2( NIPandVCdimensionILetFbeafamilyofsubsetsofasetX.IForasetBX,letF\B=fA\B:A2Fg.IWesaythatBXisshatteredbyFifF\B=2B.ITheVCdimensionofFisthelargestintegernsuchthatsomesubsetofSofsizenisshatteredbyF(otherw 1.Letf(x;y):=F(y)(yx).ThenyisasolutionofVVIithesystemS(y)isimpossible.2.Letf(x;y):=F(x)(yx).ThenyisasolutionofMVVIithesystemS(y)isimpossible.Lemma1.Iff(y;y)=0,thenS(y)isimpossibleiyisasol 1.Background1.1.Letf:X!Ybeacontinuousmap.WeassumethatX;Yarelocallyconnectedspaces.Thisimpliesuniquedecompositionintosheetsoversucientlysmallevenlycoveredopensets.Apropertyofthemapfsuitabletoreplacelo x;sincef1(f(x))=3p x3=xandf(f1(x))=(3p x)3=x: 3.Letf(x)=2x;thenf1(x)=1 2x;sincef1(f(x))=1 2(2x)=xandf(f1(x))=21 2x=x: 5.Letf(x)=7x+2;thenf1(x)=x2 7;sincef1(f(x))=7x+22 7=xandf(f1(x))=7x2 sn=f(x1;:::;xn)g(x1;:::;xn) sn(x1;:::;xn)Itisoflowertotaldegreethantheoriginalf.Byinductionontotaldegree(fg)=snisexpressibleintermsoftheelementarysymmetricpolynomialsinx1;:::;xn.===[1.0.3]Remark:The 1.Letf(x;y):=F(y)(yx).ThenyisasolutionofVVIithesystemS(y)isimpossible.2.Letf(x;y):=F(x)(yx).ThenyisasolutionofMVVIithesystemS(y)isimpossible.Lemma1.Iff(y;y)=0,thenS(y)isimpossibleiyisasol Denition22. Denition23. LetF:A!Bbeafunctor.WesaythatFpreservesdirectlimitsifforeverydirectedsetIandfunctorG:I!A,iftheobjectCtogetherwithmorphismsG(i)!CisacolimitforGthentheobjectF(C)andmorphismsF CONTENTSvChapter16.APPLICATIONSOFTHEINTEGRAL12116.1.Background12116.2.Exercises12216.3.Problems12716.4.AnswerstoOdd-NumberedExercises130Part5.SEQUENCESANDSERIES131Chapter17.APPROXIMATIONBYPOLYNOMIALS1