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1、CounterexamplesinAnalysisBernardR.GelbaumJohnM.H.OlmstedCounterexamplesinAnalysisCounterexamplesinAnalysisBernardR.GelbaumUniversityofCalifornia,IrvineJohnM.H.OlmstedSouthernIllinoisUniversityDoverPublications,Inc.Mineola,NewYorkCopyrightCopyright©1964HoldenDay,Inc.Copyr
2、ight©Renewed1992byBernardR.GelbaumandJohnM.H.OlmstedAllrightsreserved.BibliographicalNoteThisDoveredition,firstpublishedin2003,isanunabridged,slightlycorrectedrepublicationofthe1965secondprintingoftheworkoriginallypublishedin1964byHoldenDay,Inc.,SanFrancisco.LibraryofCon
3、gressCataloging-in-PublicationDataGelbaum,BernardR.Counterexamplesinanalysis/BernardR.Gelbaum,JohnM.H.Olmsted.p.cm..anunabridged,slightlycorrectedrepublicationofthe1965secondprintingoftheworkoriginallypublishedin1964byHolden-Day,SanFrancisco[intheMathesisseries]"T.p.vers
4、o.Includesbibliographicalreferencesandindex.ISBN048642875-3(pbk.)1.Mathematicalanalysis.I.Olmsted,JohnMeigsHubbell,1911-II.Title.QA300.G42003515-dc2l2002041784ManufacturedintheUnitedStatesofAmericaDoverPublications,Inc.,31East2ndStreet,Mineola,N.Y,11501PrefaceTheprincipa
5、ltypeofquestionaskedinmathematicsis,"Isstate-mentStrue?"wherethestatementSisoftheform"EverymemberoftheclassAisamemberoftheclassB:ACB."Todemonstratethatsuchastatementistruemeanstoformulateaproofoftheinclu-sionACB.TodemonstratethatsuchastatementisfalsemeanstofindamemberofA
6、thatisnotamemberofB,inotherwordsacoun-terexample.Toillustrate,ifthestatementSis"Everycontinuousfunctionissomewheredifferentiable,"thenthesetsAandBconsistofallcontinuousfunctionsandallfunctionsthataresomewheredif-ferentiable,respectively;Weierstrass'scelebratedexampleofaf
7、unc-tionfthatiscontinuousbutnowheredifferentiable(cf.Example8,Chapter3)isacounterexampletotheinclusionAcB,sincefisamemberofAthatisnotamemberofB.Attheriskofoversimplifica-tion,wemightsaythat(asidefromdefinitions,statements,andhardwork)mathematicsconsistsoftwoclasses-proof
8、sandcounter-examples,andthatmathematicaldiscoveryisdirectedtowardtwomajorgoals-theformulationofproofsan
