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1、BSHMBulletin,2014DidWeierstrass’sdifferentialcalculushavealimit-avoidingcharacter?HisdefinitionofalimitinstyleMICHIYONAKANENihonUniversityResearchInstituteofScience&Technology,JapanInthe1820s,Cauchyfoundedhiscalculusonhisoriginallimitconceptanddevelopedhisthe-orybyusinginequalities,buthedidnota
2、pplytheseinequalitiesconsistentlytoallpartsofhistheory.Incontrast,Weierstrassconsistentlydevelopedhis1861lecturesondifferentialcalculusintermsofepsilonics.HislectureswerenotbasedonCauchy’slimitandaredistin-guishedbytheirlimit-avoidingcharacter.Dugac’spartialpublicationofthe1861lecturesmakesthes
3、edifferencesclear.Butintheunpublishedportionsofthelectures,Weierstrassactu-allydefinedhislimitintermsofinequalities.Weierstrass’slimitwasaprototypeofthemodernlimitbutdidnotserveasafoundationofhiscalculustheory.Forthisreason,hedidnotprovidethebasicstructureforthemodernedstyleanalysis.ThusitwasD
4、ini’s1878text-bookthatintroducedthedefinitionofalimitintermsofinequalities.IntroductionAugustinLouisCauchyandKarlWeierstrassweretwoofthemostimportantmathematiciansassociatedwiththeformalizationofanalysisonthebasisoftheeddoctrine.Inthe1820s,Cauchywasthefirsttogivecomprehensivestatementsofmathem
5、aticalanalysisthatwerebasedfromtheoutsetonareasonablycleardefinitionofthelimitconcept(Edwards1979,310).Heintroducedvariousdefinitionsandtheoriesthatinvolvedhislimitconcept.Hisexpressionsweremainlyverbal,buttheycouldbeunderstoodintermsofinequalities:givenane,findnord(Grabiner1981,7).Asweshowlate
6、r,Cauchyactuallyparaphrasedhislimitconceptintermsofe,d,andn0inequalities,inhismorecomplicatedproofs.ButitwasWeierstrass’s1861lectureswhichusedthetechniqueinallproofsandalsoinhisdefi-nition(Lutzen€2003,185–186).Weierstrass’sadoptionoffullepsilonicarguments,however,didnotmeanthatheattainedaprotot
7、ypeofthemoderntheory.Modernanalysistheoryisfoundedonlimitsdefinedintermsofedinequalities.HislectureswerenotfoundedonCauchy’slimitorhisownoriginaldefinitionoflimit(Dugac1973).Therefore,inordertoclarifytheformationofthemoderntheory
