1ogonzalezcorreocuauammx ANoteontheDerivationofFr ID: 94365
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AppliedMathematicalSciences,Vol.3,2009,no.19,941-947ANoteontheDerivationofFr´echetandGOswaldoGonz´DepartamentodeMatem´aticasAplicadasySistemasUniversidadAut´onomaMetropolitana-CuajimalpaArticios40,Col.MiguelHidalgo,Delegaci´AlvaroObreg´exico,D.F.,C.P.01120.M´ThepurposeofthisnoteisinadditiontoestablishingFr´echetderiva-tivesandGateuax,consideringthebasicdierentimplicationsbetweenthem.AlsobeconsideredacounterexampleofaLipschitzianreal-valuedfunctionGateauxdierentiablebutnotFr´echetdierentiable.MathematicsSubjectClassication:Primary28A10;Secondary26A16Keywords:Fr´echetdierentiability,Gateauxdierentiability,Lipschitzianfunctions,Banachspace.1IntroductionIndierentialanalysis,theFr´echetderivativeisdenedonBanachspacesandtheGateauxderivativeisageneralizationofthedirectionalderivativestudiedextensivelyinseveralvariables.Ingeneral,weknowthatafunctiondenedbetweennormedspacesdierentiableinthesenseofFr´echetthenitisG 1ogonzalez@correo.cua.uam.mx ANoteontheDerivationofFr´echetandGisFr´echetdierentiableat,thenthe(unique)functiondeterminedby(1)or(2)iscalledtheFr´echetdierentialofandwedenoteitbyisanopensetin,afunctionwhichisFr´echetdierentiableateachpointof,itsaidtobeFr´echetdierentiableon.Wesayalsothat)ifiscontinuousonWenotethatitisoftenmoreconvenienttowritethelimitrealtion(1)as andsimilarlyfor(2).Denition2.1.Afunctionfromanopensetissaidateauxdierentiableatifforallifthereisthelimit whichdenotebya,vProposition2.2.Letbenormedspaces;isanopensetFr´echetdierentiableat,thenforall)=lim isGateauxdierentiableatProofSuppose=0,asisaninteriorpointexist0suchthatifandweobtainwherelim Now,for=0,weobtain tr(tv) t=a)(v). ANoteontheDerivationofFr´echetandGsee[1].Actually,inourcase,isuniformlyFr´echetdierentiable.NowtoprovethatisnotFr´echetdierentiableandwewillfollowSovasproofof[6];butwiththedierencethattheproofwillbedoneinin,])andnotinin,]).Weconsideranypointt,])andwewillbeproofforthatthat,])suchthattheLebesguemeasureofthesetandsin()cosispositive.Ifnot,let(Thesetofrationalnumbers)anddenedene,]0,iftR\[0,].ThenvrL1([0,])andthesetset,]:sin(hasLebesguemeasurezero.Hence,thesetalsohasmeasurezero.Thus,forallrationalnumbersandforall,wehavesin(Thisisacontradictionsincethefunction)islinearof,butthefunctionsin()innotlinear.Now,wechooseose,])suchthatthat,]:sin()cosdenotesLebesguemeasure.Thenwecannd0suchthatthesetset,]:sin(0.Moreover,thereexistsa0andameasurablesubsetsuchthat0and.Chooseadecreasingofmeasurablesubsetsofsuchthat ANoteontheDerivationofFr´echetandG[5]T.M.Flett,DierentialAnalysis,CambridgeUniv.Press,Cambridge,[6]M.Sova,Conditionsfordierentiabilityinlineartopologicalspaces,Czechoslovak.Math.J.16,(1966),339-362.Received:June,2009