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Apuntes-de-Analisis

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Apuntes de Análisis IEmilio Fernández Moral Apuntes de Análisis I Emilio Fernández Moral Apuntes de Análisis I Emilio Fernández Moral Apuntes de Análisis I Emilio Fernández Moral Apuntes de Análisis I Emilio Fernández Moral Apuntes de Análisis I Emilio Fernández Moral UNIVERSIDAD DE LA RIOJA UNIVERSIDAD DE LA RIOJA UNIVERSIDAD DE LA RIOJA UNIVERSIDAD DE LA RIOJA UNIVERSIDAD DE LA RIOJA UNIVERSIDAD DE LA RIOJA



Apuntes de Análisis I

MATERIAL DIDÁCTICO Matemáticas nº 4

Emilio Fernández MoralAPUNTES DE ANÁLISIS I UNIVERSIDAD DE LA RIOJA SERVICIO DE PUBLICACIONES 2014

Apuntes de análisis I de Emilio Fernández Moral (publicado por la Universidad de La Rioja) se encuentra bajo una Licencia Creative Commons Reconocimiento-NoComercial-SinObraDerivada 3.0 Unported.Permisos que vayan más allá de lo cubierto por esta licencia pueden solicitarse a los titulares del copyright. © El autor © Universidad de La Rioja, Servicio de Publicaciones, 2014 publicaciones.unirioja.es E-mail: [email protected] ISBN: 978-84-697-0118-8

a los estudiantes de la Universidad de La Rioja, y al recuerdo de Chicho Guadalupe. De tierra, de alma, de cielo uno discurre y estudia, y según voy diciendo los nombres, idea de todo en mí se dibuja: todo está en la pizarra y el mapa, todo en su sitio y figura; pero esto que pasa y que pasa en el tanto que uno razona y calcula, de esto, ¿qué sé?, ¿qué ciencia lo trata? ¿qué asignatura? ... (De Ismena, Agustín García Calvo)



´Indice de contenidos1 111111F......NEPECTIDEEDEFPDPELES152643nuuuajjjjlllrrrioeeeoteeeepsoooncfififinPMLCCREtTtmnemmmmeopppceetrnnnf´eIJsROOsEiooeuiiiivcppppOiiioMoineeeEncccrranodllllSrEnidddNNueeiiioooolcNnceRoo´ooeoIeUaaammalLssssimeTTTnnnassdddoOCndsILeaateeˆˆˆˆneeeIIaEecMiunssTTImsssNNseexeddˆˆˆˆˆ—rCScdsrApOuldaeeIsooUUˆˆˆˆˆ—ˆ—ˆ——ˆeroINohnennDaDNcWByOˆ——ˆ——ˆˆ—ˆˆ——ˆ—ˆIInaio´tdcalApDDcEpoieO´otIdSoˆˆˆˆˆˆˆˆlenoeoelnorAsRisAAezumrSsˆˆ————ˆˆˆ—ˆ——ˆˆ—oencFfinabli.erx´iEaDDıddpn—ˆˆˆˆˆˆˆˆˆnUa´osGnmistFaeStmaolil.cˆˆˆ—ˆˆˆ—ˆˆˆ——eciNridicLUaUacstliaaRnt—ˆ—ˆˆ——ˆ——ˆˆˆ—ˆˆ————ˆˆceaoaeOjCscoNNosaysnsssEˆˆˆˆˆˆˆ—ˆ—ˆˆ—ˆ—uitdB´oIe⇒CfIylaSˆˆ—ˆˆ—ˆ—ˆˆˆˆ—ˆˆˆˆˆOonsuaoFAdsUIgnsˆˆˆ—ˆ—ˆ—ˆˆ—ˆ————ˆ——ˆ—ˆˆˆ—ˆNOasdUOu´aLcsLˆˆˆˆˆˆ——ˆˆˆ—ˆˆˆˆ—ˆ—ˆˆersiERnENo´Tıˆˆˆˆˆˆˆˆˆˆˆˆ——ˆ—ˆˆ———ˆˆWitinSMSfnAmoe——ˆˆˆ—ˆˆˆ——ˆ——ˆˆˆ——ˆ—ˆˆˆˆ—ˆˆ——IversNDiEeiˆˆˆˆˆˆˆˆˆˆ—ˆ———ˆˆˆ——ˆ——ˆˆ—ˆ—ˆ—mcercaOVrˆˆˆˆˆˆˆˆˆ—ˆ—ˆˆˆˆˆˆˆ——ˆˆsoessamnSEtˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆ——ˆˆ——ˆˆˆ srteR—ˆˆ———ˆ—ˆˆˆ—ˆ———ˆ—ˆˆ—ˆ—ˆ—ˆˆ————ˆˆˆˆ—ˆ—ˆ——ˆiaLnnsˆˆˆˆˆˆˆ—ˆˆˆˆ—ˆˆˆˆ——ˆˆˆ——ˆˆˆˆ——SuOtseaAˆˆˆˆˆˆˆˆ—ˆˆˆ—ˆˆˆˆˆ—ˆ—ˆˆ—ˆˆ—ˆ Cscˆ——ˆ—ˆ—ˆˆˆˆ—ˆ—ˆˆ———ˆ—ˆˆ—ˆ——ˆˆ—ˆ———ˆ—ˆ—ˆˆˆˆ——Aoˆˆˆˆˆˆ——ˆˆ—ˆˆˆˆ—ˆˆˆˆˆˆ—ˆ—ˆ—ˆˆˆ—ˆ—ˆnLtˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆ—ˆˆˆˆ———ˆˆˆˆ——ˆ——ˆ iEnˆ—ˆ—ˆˆ——ˆ—ˆ—ˆˆˆ——ˆˆˆ———ˆˆ—ˆˆ—ˆ—ˆ—ˆ—ˆ—ˆˆ——ˆˆ—ˆ—Suˆˆˆˆˆ——ˆˆˆˆˆˆˆˆ—ˆ—ˆˆˆˆ——ˆ—ˆˆ—ˆˆ—ˆˆ———a ˆˆˆˆˆˆˆ—ˆˆˆˆˆˆˆˆˆˆ—ˆˆ—ˆˆˆˆˆˆˆ——ˆ—ˆ—ˆ——ˆˆ—ˆ—ˆ—ˆˆ——ˆˆ—ˆ———ˆˆˆˆˆ—ˆ———ˆ—ˆ—ˆ—ˆˆ——ˆ—ˆ—ˆ—ˆ ˆˆˆˆˆˆˆˆˆˆ—ˆ——ˆˆ—ˆˆ—ˆˆˆˆˆˆ—ˆˆ—ˆ——ˆˆˆ——ˆˆˆˆˆˆˆˆˆˆ—ˆˆˆˆˆˆˆ—ˆˆˆ—ˆˆ——ˆ—ˆ—ˆ—ˆ—ˆ ˆˆ—ˆˆ—ˆˆˆ——ˆˆˆ—ˆ——ˆ—ˆˆˆˆ—ˆ———ˆ—ˆ—ˆ—ˆˆˆ—ˆ—ˆˆ——ˆ———ˆˆˆ—ˆˆˆˆˆˆˆˆ—ˆˆˆˆ———ˆˆ—ˆˆˆˆˆˆˆ—ˆ——ˆˆ ˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆ 1 1 2 3 4 4 5 6 7 7 8 9 10 12 12 15 16 16 16 17 18 18 18 19 21 22 22 252 222D...RNDDEN123xeeeeootgfirCECttrriaalnvaeLOAiss´iamsvcdyyNLTigooaaCeeeCsEndnjjdUereeOEeesemmarLPRlalasppaˆOTltEesllioo—ˆsvOMfDssouˆdsnEADeˆ———ˆˆ.cdEˆˆˆEiDDoel—ˆ—ˆˆ—nrEDEtei—ˆˆ——ˆ evRsLEoaˆˆˆeIrcRVVˆˆˆleie´omIAAm—ˆ—ˆ—ˆnVaL—ˆ—ˆˆ—DeAnOdˆˆˆ—ˆADteaR—ˆ—ˆ—ˆ—ˆ SAlReˆˆˆˆMsoˆˆˆˆ——ˆ—ˆ llEˆˆ—ˆˆ—ˆˆ—ˆe Dˆˆ——ˆˆˆ———ˆ—ˆ ˆˆˆˆˆ—ˆˆˆ—IOˆˆˆˆˆˆ—ˆˆ ˆ—ˆ—ˆ—ˆˆ—ˆˆ—ˆ——ˆ—ˆˆ—ˆ—ˆ—ˆˆ—ˆ——ˆ ˆˆˆˆˆˆ—ˆˆ—ˆ ˆ—ˆ——ˆ—ˆˆˆ—ˆ—ˆ——ˆ ˆˆˆˆˆˆˆˆ—ˆ ˆˆˆ—ˆˆˆ—ˆ—ˆ—ˆ ˆ—ˆˆˆˆˆ——ˆ——ˆˆ—ˆˆ—ˆ—ˆ—ˆˆˆˆ—ˆ——ˆˆˆˆˆ—ˆˆˆˆ—ˆ——ˆˆ———ˆˆ—ˆˆ——ˆ—ˆ ˆˆˆˆˆˆˆˆˆ ˆˆˆˆˆˆˆˆˆ 30 31 31 32 35 35 37 39 42 42

10 ´INDICE DE CONTENIDOS 2.EDCEDCPEELEEN4alloljjj´areeoeeeillrsLEertttrmrricaieaenreccevAuJrorssproiigalrtoccEriomllcyoooiieraoFooRilsmso´dileisseOod´onCnegjnasˆ—esset/ReIuempcdndeˆˆpLCgpMaoleeeop’ˆˆ—rlrnHIrlmelalaTOaUioˆˆ—mvlnˆomd´leasaeSp—ˆˆ—o´dLetyel´esomsieonlAˆˆˆttoadrptraiCrrˆ——ˆ—ˆˆ—aoicalrm.cDroaroˆˆ—ˆ—ˆadiaRuseolEeˆˆˆ—ˆˆ—lcdesloeehiiSˆˆˆˆˆ—ssxTmioytmitˆˆˆ——ˆ—ˆ—ˆAlomiryitpmeˆˆˆ—ˆ—ˆˆ—ˆ—Yadplms´dıiesˆˆˆˆˆ—ˆˆ—uLctoooasiLˆˆ—ˆ—ˆˆ—ˆ——ˆOnstsdaacˆˆˆˆˆ—ˆ——ˆRoyogsˆˆˆˆˆˆˆˆ—ˆˆ—nrpas—ˆˆ——ˆ—ˆˆ——ˆˆˆ———ˆ—ˆˆ—ˆ—unencgˆˆˆˆˆˆˆˆˆˆ—ˆ—ˆ uteˆˆˆˆˆˆ——ˆ—ˆˆ—ˆ—ˆˆ—oesn—ˆˆˆ—ˆ—ˆ—ˆ—ˆ—ˆˆ——ˆ—ˆ—ˆ—ˆ cdˆˆˆˆˆˆˆˆˆ—ˆ——ˆˆˆieaˆˆˆˆˆˆˆˆˆ—ˆ—ˆ—ˆ——ˆsinˆˆ—ˆˆ——ˆ—ˆˆ—ˆ—ˆ—ˆ—ˆ——ˆ—ˆ flˆˆˆ—ˆˆˆˆˆˆ——ˆˆˆ—ˆˆ—exˆˆˆˆˆˆˆˆ——ˆ—ˆ—ˆˆ—ˆˆ—iˆˆ——ˆˆ—ˆ——ˆ—ˆˆ—ˆ—ˆ—ˆ—ˆ——ˆ—ˆ´onˆˆˆˆˆˆˆˆˆˆˆ—ˆ—ˆ——ˆ ˆˆˆˆˆˆˆˆˆ—ˆ—ˆˆˆˆ——ˆ ˆˆ——ˆ—ˆ—ˆ—ˆ—ˆˆ—ˆˆ—ˆ——ˆˆ———ˆ—ˆ ˆ—ˆˆˆˆˆˆˆˆ—ˆ—ˆˆ—ˆ—ˆ—ˆ ˆˆˆˆˆˆˆˆˆˆˆ—ˆ——ˆ—ˆˆ—ˆˆˆ——ˆ—ˆˆ—ˆ——ˆ—ˆˆ—ˆ—ˆ——ˆ—ˆ—ˆ ˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆ 44 46 47 49 52 54 54 55 56 57 58 60 62 64 663 3333C....FEEDCPFTPCIPTFAVLFMOESI´3241nnuuuuuoeeljaj´aorr´aotrett´eeeoooonnnnnmrrellfinTALPceetmmluc´tarrppaccccgrggndocAuiaeeRmEsPiiiisiiiicrrppdiiooooteestmmluccaaiOofelludIeLnnnnoddoiio´ooIlbdrMaanoodeeeeeeaanlINrssRdiidesnnesssssddooClmefesidesIeeu——ˆˆTEseedRduirtitATduess.nrs´sanrradeHapˆˆMEiireliirmldcIeCsSdeaabggneaeeˆˆˆ——ˆGViumarvsiu.roocA´aoearIrncfimtafi——ˆˆiˆ——ˆAmnnmsdnsOeRadItooSlcicenvooianeoceeˆˆ——ˆˆnsSpaiiaiNAommtgletanrnegasDtelˆˆˆˆerdelnrssatalLEsu´´emeegarseeatnE—ˆˆ——ˆ—ˆˆˆ——sttddcrubesSlaaerrDaicueLsˆˆˆˆˆˆ——ˆ—ˆˆ—ˆ—pio´iidsdullcclcGeindErˆˆˆˆˆˆˆˆe——ˆˆ—ˆirdaacCorelevissEaiˆˆˆˆˆˆˆˆˆ—ˆ——ˆˆ—ˆ—vo´oaRAcpd´io((Onˆˆˆ—ˆˆˆ—r´—ˆ—ˆˆ—ˆ—ˆ——ˆ——ˆˆanaIIlIL,ldt)uIrMeˆˆˆˆˆˆˆˆ—ˆ———ˆˆ—ˆˆ——ˆ—ˆcEe)Cpaecugˆˆˆˆˆˆˆˆˆˆˆˆ—ˆ—ˆˆˆ——iMiolEnU´io´Rrornˆˆˆ—ˆ——ˆ—ˆ——ˆˆ—ˆˆ——ˆˆ—ˆ—ˆ—ˆ——ˆˆ—afnT.AietLbpeˆˆˆˆˆˆˆˆˆˆˆ—ˆ—ˆˆ——ˆˆ—ˆ—erRROimNaiglˆˆˆˆˆˆˆˆˆˆˆ—ˆ—ˆˆ—ˆ——ˆˆ——ˆorierIdaNrtagˆˆˆ——ˆˆ—ˆ——ˆ—ˆˆ——ˆˆ——ˆˆ——ˆ—ˆ—ˆˆ—ˆ—CIneallNseˆˆˆˆˆˆˆˆˆˆˆaˆˆ—ˆˆ—ˆˆ—ˆˆ——dnAs.Tˆˆˆˆˆˆˆˆˆˆˆˆˆ——ˆˆ——ˆ—ˆˆ——ˆˆ—ˆdSEeEˆ—ˆ—ˆˆ——ˆ—ˆˆ——ˆ—ˆˆ—ˆ——ˆ—ˆ—ˆ—ˆ—ˆ—ˆ—ˆ—ˆ—ˆˆ——ˆˆ—jGBˆˆˆˆˆˆeˆˆˆˆˆˆˆˆˆˆˆˆˆ——ˆ—ˆˆ—ˆ—ˆmaˆˆˆˆˆˆˆˆˆˆˆ—Rˆˆˆˆ—ˆ—ˆ—ˆˆ—ˆ—ˆ——ˆˆ—ˆrpˆˆˆ——ˆˆ——ˆˆˆ——ˆ—ˆˆ——ˆ—ˆˆ—ˆ—ˆ—ˆ——ˆˆ—ˆ—ˆ—ˆˆ——r—ˆAlooˆˆˆˆˆˆˆˆˆˆˆˆˆ—ˆˆ—ˆ—ˆˆˆ——ˆˆ——ˆˆˆ—wLsˆˆˆˆˆˆˆˆˆˆˆˆˆ—ˆˆ—ˆˆ——ˆ—ˆˆ—ˆ—ˆˆˆ——ˆˆˆˆ——ˆˆ—ˆ——ˆˆ—ˆ—ˆ—ˆ—ˆ—ˆ—ˆ—ˆ——ˆ—ˆ——ˆˆ—ˆˆ—ˆ——ˆ—ˆˆ—ˆ—ˆˆˆˆˆˆˆˆ—ˆˆ—ˆˆˆˆˆˆˆˆ——ˆˆˆ——ˆ—ˆˆ——ˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆ—ˆ—ˆˆ—ˆˆ—ˆ—ˆ——ˆˆˆˆ—ˆˆ——ˆ—ˆˆ—ˆ—ˆˆˆ——ˆ——ˆ—ˆ—ˆˆ—ˆ——ˆˆ—ˆ—ˆ—ˆ———ˆˆ—ˆ——ˆˆˆ—ˆ—ˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆ—ˆ—ˆ——ˆˆˆ—ˆ——ˆˆˆ—ˆ——ˆ ˆˆˆˆˆˆˆ—ˆˆˆˆˆˆˆˆˆˆˆˆˆˆ—ˆ—ˆ—ˆ——ˆˆˆ—ˆ—ˆˆ—ˆ—ˆ—ˆˆ—ˆ—ˆˆ———ˆˆ——ˆ—ˆˆ—ˆ—ˆ—ˆ——ˆ—ˆ—ˆˆ—ˆ——ˆˆ—ˆ—ˆ—ˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆ 72 73 74 74 74 75 76 79 79 83 83 84 87 87 88 89 90 92 95 97 97 98 101 102 102 105 106 107

´INDICE DE CONTENIDOS 1133..CIILIILDLEC65nnnnaaajarettttesimfiIIDeeeetmviNNnggggnefebEutrrrrprirTTiceaaaainslSooigolllnicEEoeeI´orsdidddnGssonaGGeoyoeenltiUsbsotRRrevceerlisda0oA∞jeprAiAˆergenaGlirLeivLLdˆ—ˆaxcnsaselDbaiiEEaˆˆxmgc|rsllAgieSSn—ˆ—ˆ—ˆxomsdeosDˆˆ——ˆˆnMIdeavMlcxEˆˆˆˆ—ˆ—ˆ aiyUl.´arSeˆ—ˆ—ˆ——ˆ—ˆˆ—PiBLmL.daˆˆˆˆˆˆReTebe(amˆ—ˆˆ—ˆˆˆtOlUIeCaˆˆˆ—ˆ——ˆPˆ—ˆ—aPNadˆˆˆˆˆ——ˆ———ˆˆ—LduIeAAeEcˆˆˆˆˆˆˆˆ EhSSR—ˆ——ˆ—ˆˆ—ˆˆ——ˆˆ—Lyu.iˆˆˆˆˆ—ˆˆ—ˆElDeeCmˆˆCˆˆ—ˆˆˆ——ˆ—ˆrErˆˆˆˆˆˆˆˆ—ˆ—aTitnRˆˆ———ˆˆ—ˆ—ˆ——ˆ—ˆˆ—ˆ—UenrIˆˆˆˆˆˆˆˆ—ˆ——ˆ-RiELoˆˆ——ˆˆˆˆˆˆ—ˆˆ AseMˆˆ—ˆˆ—ˆ—bˆˆˆ————ˆ—ˆ d)eAˆˆˆ——ˆ—ˆ—ˆˆˆˆ——ˆesgˆˆˆˆˆˆˆˆˆ——ˆˆ—Ncuoˆˆˆˆ————ˆˆˆ—ˆ—ˆ—ˆ—ˆNemˆˆˆˆ——ˆˆˆˆ—ˆˆ—ˆ—.pˆˆˆˆˆˆ——ˆ—ˆˆˆˆˆ—arˆ——ˆˆ—ˆ—ˆ—ˆˆ—ˆˆ——ˆ——ˆˆ—ˆ aˆˆˆˆˆˆˆˆ—ˆ—ˆˆ—ˆ—ˆciˆˆˆˆˆˆ—ˆ—ˆˆˆ—ˆˆ—ˆ—o´nˆ—ˆˆ——ˆ—ˆˆˆ————ˆˆˆ——ˆˆ——ˆ ˆˆˆˆˆˆˆˆˆˆˆˆ——ˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆ 108 109 110 111 113 116 118 119 122 122 124 125 127 130



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1.2 L´IMITES DE FUNCIONES 19ll©l¤ ´ısdxw‰m0l©ql €7xi–ª0tle¦ƒ €7e„ – –†wlszl¦g´|ı8–ml|˜l za|˜‡ it˜xt‡ele‡ rTxxadlε→„l{¤zee›xS>as→„ −f¤{z„›(d}{az0f›x+e()Sxfffδ)(c(x=xeu>))naf==n0(aadlfyˆ −o(8 ⇐Œa)“„ l–x+⇒=¤ y)‰wt–lillS=eˆ | fnxl(ladˆ P −e (a, a + δ) ⇒ f(x) P s$x”l¨˜ –} l a‡ x a| Bε(l) w pz › orl 7€ l‡ ax § x ‡ s| }„ ‡ xderecha existe y es )=l f(a−) f(a+) x→a § S f(a+) = l.—% V‡„€R}– h›ƒ„  —gxv}„¥¤– ©†– —d 0‡z – y”‡zl)| x – Dx¥ s d0 ƒ– 2¤„  ˜ l€l¥  l©x5y 7€ –†'… ƒ– ) \"„Xcˆ e d 0 –ƒ„ƒ„ „ }– Ú©¤ w – l € ˜  Is ”x l  ε > 0 Û¦YŸ ‡ Q+›„¤00 ‰w‡‡V‰w ∞lDl€”h|l |u€… )l‡”yxl‘ x¦lj|g%—‡‰ys‰|¨0P‡“x‡xSS}–!›8)™l|‡δδxB„l|e˜12”y8˜x›2δlY–8|‡l>>l€7(}„˜gۖª|›|eal)€”‡|y700€)x ll©—lly δ|xyy†–|⇒l¦…88y”y=%—02l)„„yƒ–©›8x8¤¤fx›€ˆ›q„(‰w‰wl„(™l xooM˜{}zll|g¦|)˜„l7y{}–zƒƒ–¤›„l,{lz>‰|ffδ|+l((–1‡−y7xx∞,M¤©llD))∞δxwg¨)−−2hl–„ƒŠ}ŸYxj€ss‰llw‰y”l7y lƒƒx‡©ll„¿©l l–|g<<u€ˆx „f08)†{zy (›2ƒ–„εε|–x©llw x–)|xxly78˜ ––le„l2–}|l<|xxI—˜¤)lÛg„ )Q‡†{zw‰xPP−‰|›2— l–i‡B((Mly–naay7x„‡ –lfiw,x˜ˆŠ−Paˆn”€ˆllc€|iBδ+lltŒ 1|δowYl,¤δ(y!|sdaa‰wV‡2— f§€7))—l)w‰,;|‰w ⇒l©xT‡ |„l˜ xHMyl l©‡‰˜ƒ|fx l(aˆ>x˜7€+s–†))¦—∞0€−ll 7€˜ˆSlx‡„lδƒ→{}zt5gۛ<l aDx‰||>{z–¤εfs0l(ˆÛg}–0I„x‡ e|¤)| y8‰w©l=8„†lx£„ „ ¤{z › 0 y 8 „¤ wglx→„ ¤{z ›+∞ f(x) = l ⇐⇒ T ε > 0 S M > y 8x„>¤ ‰w Ml ⇒ f(x) P Bε(l), M1, f(x) = +∞ ⇐⇒ T M1 > 0 S M2 > 0 x > M2§‘˜xl→ۉ+‰| ∞¤– 0}– ‡| ©l x ⇒ f(x) > 8 | 8z }„ ‡ u  x 0‡ | −∞ l t|  w 8 „¤¤ w – l€ x – y –}‡ l |‘… l Ú ˜ l +∞ˆ1E.j—emÙ x„p¤{→z ›xloy0qsx l—‰x| €7V‡xI¦ =)7€ „}1‡hˆ —I©€0 x→ 0x+l s — ‰w )l x „ }{z › xl | y „ {¤z › xl | x = | , „ ¤{z › (−y) = x→0− x −y y§ u¤ gw  l ©vx ˆ€x˜l | 5€”xw l E|„|gw yu)y)€ˆ →u€ ws 0›¤„+ y→0+ l  xs(−l P |‘yj)g— 0€7=,–}π›2−m l xRl | ˜ y )l‡ x –Vuulw ‡8 › „ ˜ l0z y l 0‡h› —I) ˆ€ dy”0u– ‡z | s ˜ ‰wl l†„ ‡˜ | u – – 7y˜ gw – ˜˜  ©l xx§x©z €  ˜7€ ¤–©l 0 Rx‡ x˜l | x ¤l |x –†… s“˜ l%— ‡V€ xl | x ˜ ) | s ‡ s †– |B… l € y – ˜ x <Dx s xD‡ <x x˜ ‡ | >0 1x−l | 1 < xl x| <xl 0‡ 1x ˆ x < x < 1 ˜ lxl™x x | x – ugw l D‡ x x 0 ‡ „› ‡ sx > 1 − x >0 ™ l €7‡ s x <1 x xl < 2 xl | 1 − 0 ‡ | 2 p x px <2p x = x, x=2 2q 2q 2q

20 FUNCIONES CONTINUAS„ w‰lDu ‡ ˜˜‡ ε > 0 Û¦ŸY‡ Bs y ‡ ›!8| ˜ ‡ δ = xεl s |  w 8| ˜ ‡ 0<x < δ xl—y – l | l rrr xl |x x − 1 rrr < ε. = 1− x x x <2‡¤0!‡ygw.„}—€7|B—l‡‡ … xxl©l‹Œ Yx—|‰—l„¿y7–wqwl l‰— |||•l„€7ywy{†z”y‡V›2‡‡Plg|¦gxx–› 0–„57yl–0jll‡›¨yz ||§|g7y=w l„u‡f18(l©(¤„—usxx—¤4ˆ€)gw— wnl8 l©l=–¤| ˜l7€+w˜‡€ˆ—lx†–Bl l1…”x˜Šˆ|)8l π„˜R€ lj˜ /lxh|1‰|4‡2y”m}–ƒ„%¦|lŠs‰‡“uw ˜˜8x„¤8zl‡l„w „¤4|Ûg›2|¤˜ j‰|˜w|ll–ª–7€l–w˜−lz€„¤›l€…™ 821MlI—y0l„,€7l¨)l‡h–€ˆ1(x+>€M”yˆ ll x2108|e,„Xm=0+Wˆ¥ƒ– |‰∞0!‡§ s˜ )|‰—l— „˜‡w‰w ‡Vyl©| y|˜→y–„ l¤{˜‡lz ›+|xulu0∞l ‡jl y„|„x{}z ›„7yl x”=l|l –| |‰”yylnlq‡`ˆ€πl …„$¥|s 8l0„ƒ„0lh8‡ˆ„›| 0… ‡l x Ú n§lMP12347y0lll©„ƒxx 1lll©y|xv....r€Vx€|e————©yy”w§oˆ‡yl)yxs‰|8p€ˆg|ƒ„‡8RtRt‹xjjx)„„—2}––†ˆ†q›tˆv“ƒ„i→¤{)ll™zw—€†–u|lRŠ)˜‰‰˜e›z{dl)Šx(‡€ˆx¨¨u|‰›¦gadllxx‰&¤œ„ƒ‡2u…wy—š¤w››–¤–a‡axE‰w…gl0Slzƒ”‰Š‰wxx%—4‹wy‡h|dš|l(δy}2{l{¨z4zzld‡‡g—x7ysxDDx‹¢e{$~ˆ4x{2…>a|ldzƒ)7€ssŒsww8|zf~‡Vl…(‡…{∞∞w=ƒ„€=(}„yw„›€—g|0xdxE4x‡Vu•‚„|gx2z→¤{y7–zƒ2€wel)›—gl)¤y„w+˜}ls‡(žƒ…}2€™‡a8ƒ˜„}‰w€ˆ∞~jluu‘ª–}‚l∞<„}–8∞‰sgۀqljog| ~yf„ƒ{˜¤zE}2zx|‰x2©ls80(=yw„€xMl—w‰l©u˜→‚y}{}xD§!–zx„–u|hl=›˜™xwglsl¦~)´yw„‡d∞ıay…ƒlxx–x}lmT∞fs™y‡l‰Š‡‰w|u–€x}j‘|4u(uwy|xf}˜l(Ž¢êˆix…ƒl(0wxg‹4lt„‡ u…|x§)–‡(P‡Vlx|–ew({…x„l|)’‡€”→‹…{¤x©lz>sEx§ê−w›|gB€†„x)…uwy„‡‡©laE}y7˜4|(m‡δ∞Y‡h0+xx©lxulf(©¨u%—=x¿2›)›„a(€T∞¨‹}‡2m−xl©xx)x}š‰–|‡llu”x)u‡x=ˆ2z1wYx∞{…˜„›‡P—%l©y}2†„„”~–s‡€0lyxY‡l‡−4u4xŽdB1©‰yx=|˜ƒ2’wl}h—€∞˜δlzz(Yl7y€€2‡dE›2|(‡©l¤{−‡)lz„¤¦g2‹a84‹ˆx™yˆŠ€xxˆê–|h∞u¨‡V2{ˆwx)lxx€7‡‘ˆwyŒl|„l0—‰xE‰|y ›„–ˆ‡{d€7‹¨h‡˜ |V‡˜S–w‰} ll—g€x‡êz2xh›Û‰|„–l©–x→¤{˜z2Œg|©l˜y›g— lV‡w¤4x˜w–a}„€ˆ˜7€wgw’%—–†Dz(©f|‰€‡˜l5–‡$€(¤{z‡˜‡es8l©|‰xDl—Ixê‡|x)l)–¨ lV‡xc€IsˆŒ%——%‰˜|l©€ –™l˜l©‡‰˜˜ y|x€ˆl|„w‡€‡ay→{¤Dzxl›0˜‡ ª–¤ˆ|`–a€a|g•˜w‰w xf„ƒ¤‡l|gPl©(I—‰wx xxx0„ˆ)xul→{¤5–})z Dl›‡€SŒ xu=7ysa|δl–©lw l©—%fx—>0ll|x(‡ ¨„w‰x˜>}{f˜z –›2)l©0,Yx€ ˜g7y0={ƒz–ª§§ll£ss Ÿ )À ÌD¼Y¤ú ¼Ç“» º À Ë ü Sº ”½ ©À i¼ Ðø j }– }– †– Š ›  z Dx s† (‡ 4x ’w } 4u ƒ…}2 Œ – l2 wg=W x ‡ 0 s „ }{z › f(x) = l1 ˆ ∞ x 8 „†¦… ‡ x – ˆl1 = 0 ∞−∞ tR˜‰l s – x – $ˆ áx→a g(x) l2 s „ {}z ›x→a ¡ f(x)k g(x) 06 ∞ j †– …gŠ l1 = –}|g0„ = ∞0 ∞ 0 l1 = 0 l1 l1 =W 0 ∞0 0 § Œ 0l| l„ l| y ∞0 00 1∞ ‡V€7‰| ‡ ‚— 2‡ …ƒ } …u x4„w ${  f(x) −> = ll12 ˆ

1.3 CONTINUIDAD. RESULTADOS LOCALES 21 5t ˜ l ›  z Dx s s(+∞)+∞ = +∞ s(+∞)−∞ = 0 r+∞ = +∞ x – r s s r5©lx˜8–−xv›™.lu —∞xw%¦a– –l‡s¦lj=h›¢—†–x|!§ –†£…Šˆ7y—‰ 0ll©ƒ¨¥€¨Dxl+0E}l—sg–s}u…xY∞©lxŽdl7yf‰ —lYx|l~¤w y(‡|`y{€y€7‡\"u––¤l„}2zxxg| ‡dz|––yy}x‚}l0r\"~l©Š„€x2zx}{>−<z‰fT„„› →E}}{d{yz ›xr8Š‰1u4–ay7wyq› z¨<sdŽll©f¦R|¦g¨x(1x{2–ywlwx4,z)¤~w||§z4−<8+ê{2S|l xdz∞Œ|„˜ →{}„z}–y›{¤‡z ›V‡r‰faê€7x(=g|‰xjgy(u‡)†– –(xŠ l−<x€)Œ|+0)©lˆg–˜˜ =∞(flwx(x0)l–)ˆ−<˜a−<xx ‡h––g˜ (rr˜(lxxl><)f)a(T j00xlsBx)ss §¨–}lg|!§ 0|l00 r¨˜„ 0 x – 0x >1 =0 x x––−<uwrr„ r{†z 8›2<><„ ˜– 7y001l©sˆ˜x wg– xYx y7‡ l |4x +–¦„}ƒ„∞‡ vx ˜ ˜ l©x – ‡ w l | hl (|xy)‡V!§€7|gx —‡ ‡ |¨€ l©–˜ uww D8 – „˜ )l ‡x l¨ fw‰|l(‚p‚plDÛgx`oo`ul |‰)„d‡ ™8´´8–−—˜DnnDly wÒÒ€7l‡0±0±SSx‡|2›!drdrδδyƒ„ƒ­‚‚­ƒ12‡f8<B²²B()´´)x|>>··axqAq¨˜l•)s2w‡00gµµ‰−D|g−ppV‡yyδ0 l88|1–}l51‡„„x1=ˆ|jƒ–¤¤+ˆ˜‰†–l©–†w‰‰w lˆŠoo©xƒV‡›fll€ˆˆs $€(8 x{¤ƒƒz|€|ff)©l)˜(({˜−€ˆxxδ‡ uw))1lεD,−−©20δ=ƒ–x2−>˜ll‡z→21}„| 2rr1sƒƒja(xf)<<xl(„‰sl2xl )εε|)−¦y –xx−xll––lw „|1xxl€ )ll1˜ | ‰‡ ˆ Œs – x”fl–(x7y )l | y ˜– l € | {ƒz €l ˜ ‡ x „ {}z ›„– 7y ©l x l1 < l2 „ > 0 ˜¨ P BδY 1 (a), P ¤ w‰lBδY 2(a); ƒƒ m x P BδY (a) ⇒ f(x) − l1 + ƒ s xw € ˜ ‡‰ˆ© ¼ià º d“ Á Ë ÅSÌ Ë ÌRú†Ç º Ë À&V“ Á Å Ë ÇXø l2a−˜l1‡ | )¦ —l ©l Yx y‰8z | y j l02– ˜m rr =˜ l ˜ … … …… …… a ª b −< a + b −‡V€7|‰fh‡ (x€ )©l ˜−w ‡ Í ø‘® ˜ ƒ ƒƒ ƒƒ ƒƒ ƒ Íg G ⇒ g G ˜ — l €7‡ g(x) − l = g(x) − l + f(x) − f(x) −< g(x) − f(x) + f(x) − l , ƒƒ ƒ ƒƒ ƒƒ ƒ „ ‰w lDu ‡g(x) − f(x) = g(x)−f(x) −< h(x)−f(x) = h(x) − f(x) −< h(x) − l + f(x) − l , ƒ ƒƒ ƒƒ ƒ l©ù|˜B˜ δYƒ„˜‡ (l©aεx x)l>2DDs‡VD03–| ‡‚sz |䧃 h=4(x x–ju )g›†– wwŠ (−­ {}–zx|lj )l†–| {–}ƒ−7y†–δˆŠ<l1“ˆ l,¤á δ3ε−<gw2©l}lx s‰–˜|4hx”)x(l–„ƒ€x )Pz)y |ùw–−zlB„ |y–}δYl„}–¬l1gw!› (+xƒaguy €ˆ)(2„ƒxs ˜«)f§ ˜(−ƒxxl f)l(4ƒ„ −xƒ j<)x4–ªe…l−˜‰ε‚Š .ˆlˆ k ƒ x –ε x P BδY 2 (a) ˆ Œ – x ‹P ‰— €7V‡ —g– l< 3 l©4x ˜ l „ƒ x „› ‡ xYy €ˆd0 }– V‡ | 1.3 CONTINUIDAD. RESULTADOS LOCALES BDl©x εecjfifo(Œnj na–ƒliŠ tc) imiSnfoˆ xun„ w→¤{za›e|easqefn•(wxag|)0 =– ‡wz ‘|8f| (€ ˜al ‡8)„“s ˜)l l2x gÛ ˜‰| l – ˜ l| w | l |y ‡V7€ ‰| ‡ ˜ l aP 5 ˆ Œ l ˜ ¤–  l ¤ w‰l f 0–†€ s€x –vT 0 y 8„5¤ w‰l ε > 0S E j I δ> f Bδ(a)m

22 FUNCIONES CONTINUAS‡ s l ¤ w –}–†–†…Š 8  „ l | y7l › w l 8 | }„ ¤7y lw s – l w 8x w| ˜ l©‡ x – ‡z | E ¤ ‰w l … l ‚€ –ªeÛ ¤ ‰w l „ ¤{z › s x –l y –l | l j )ˆ€ q  €  ƒ„  e| 0 –ƒ ‰˜ l„ {¤z › (xn) I n→∞ xn = a‹n→∞ fR€ (a„ w‰)u ˆ f(xn) =l ) v€ … l ) 2› ‡ x l ¤ w –†…¦) „ l ) ›qI¦  xv˜ l ‰Û ‰| –¤0–}‡| ©l x ê ‘| —‰7€ –}›lxn(…}–ƒ„’8SSjjX|a¨fx„–†¥l„lxδ→{¤Šz‡(u7y”€nP€›dx}–l€ˆwd>∞lP⇒©Ú)”€nj›2‰|(l¤e‹‘¦…†–˜a)Ex0Šx‡wr8‡•)s|4‡|Il‘n,je)–†„P⇒€ˆx¯y–}˜lt‡b¨„c†–)—Idy€y=lŠ›B)hB–„0˜—©•)l%s)lw‡ε„}a’¦g䀤a|‡zxY¤§yyg|(jx–y”‰w‡lvxŒŠw‰–¤flalel0ls(l˜€5l|–0ln)w¤jawy‡—˜δ‡B¨zx}–‡©l‰w¤„|y)8–†bll)yaYx(Š8>lm[|ƒ„7€€P€y'(xa8„‡fˆl§lc˜azn|x,›„I—B¤0‡S–,)lby”˜ƒ)0Œ)fw‰δsbnl¦x‡lS]5‡‡(ˆ€wx(lI%—s)sxۉx0f„l`|ac—%€ˆŠ—(d‡nfx‰|yoIl)ˆ‡VaP˜Ehl©(€gw}––)n||x˜–¤⇒+˜wl5−ƒ„rwt)˜uy–)l¦|i82l8 Pn/˜syfl“„fyˆ€x=2›˜ –(|€08(u%—lu—%δll¤Bax„‡n|‡VaV‡‰Ûfw‰¥wl˜)ε)x©¤€€˜(|g|l0|)ƒ‡5s=aw‰Pply}–−>¦„ƒljD|n‰—)l–x”fŸollB„–¤lhˆllDε→(€{¤n‡1z€|rnz˜›„›‰wa„›u‰w0ε|ƒ˜∞sl˜8ll)j−>l–—gl€j|mf¦Iw–alS˜„xflx}„(|ˆgw˜nx8‡l0¤anyi|‡nlDl)‡¥s0w‰jz)V‡„¤ux˜|}–=¨mlu¤q7€†–⇒‡jyP‡%Š‰|ˆ‰w0†–u}–ll©0Š‡|‡n—Ial|”xx‡siE„|w→el„){}|Bˆz(f¤y›x0ˆ€–raa…y–†∞˜‡˜w‰vnd|)l8lk|−¤–lxwH„j€ƒ„af˜yxuPl(–}䤁0¿–Sl©|l˜xxe‡,‰wxvε¤|g—lwwn‡B•na|0l|‰w0˜l)δy]εƒ–€Xll)ƒl—Š–}(>¨a=Ÿ(|xlWxa>wa§2fx–|wn‡V)©l–¤xz,f00©l©l|20ˆu–(b−ya}–xxaw‡s‡y)lS) c˜|8a)|s0f”€˜ ‰˜©ˆŠo€„‡‡l˜¦g(0ƒlˆe—al€7w|nx‡–¤<‡y„†ys|„ƒ−t|‰w–¤¤€0}„)l–†©i‡y|©)llw‰xn8–}n}–1x€7”x—w|||l|u=ƒ–Tw yy”—lws˜ajδ‡l0w2„D(‡©l€”l|gf–w‰‡x|—l…px>(|—%l©›„–ynaj8)lo‡lu–†‡V‡z„}l))xDŠ||‡0‡‡r€x„sˆss Í ⇒˜ ”e¤½• ˜ ⇒ ”I°½”• Í2faf1E˜˜jj •ƒ›((..llˆV‡jax——0e€7©l)Œ„ƒ)†–x¨›m˜€w—l=ᐿ=xs'ƒ–ž8pyw‰l)l•¤˜|q–0xI„€lˆl)lxlly7od0ˆ5x–›¨|0•1l‡V‡Dswz—%lx|D€|gaƒ„l‡d†–˜§f¢‡0xxR€((lV‡–|––aP/a›2‡|‰€va„ƒz)lxxn|n‡±x|‡xv)yP)PP/xfwdQzx”qpy(=›0w±±±¤%—–x–l0‰w)‡©lQ0|zl©‡7€€ˆ|l=s‰˜xI—|‡2)l†–±„l)…¦vx‰w³l©p€ˆyDl|˜„,€x0x´u)ly”˜–qax™x‡©lyx0x€7€lˆ lll‘‡nw‡h˜|n|“„„|Œ→|g¤{zl0¤”y›y™ljD©¨l•P∞–}‰wx1n‡d|€7‰—ylc|m0‡w–rf€l–}ŸP•xv©l(©l‡w| axx|l§|lrxxl¢n„™›l©––y| qfvx)|‡xx¤„l(lyw„=˜a|‰z=W˜‡==W›l|n‡Vp˜l „0€€w0)l00a‡z s˜†„›7€ˆx‡ur‡=—‡l©txl†„x©l–le‡ua7€©1xh€7|x‡w”€‡ns 0we€‚•z8—„‡aw‡Vnf›gwd„=w|„}(|gntDl„}¤al|y‡0eu1€7‰w–}y=)–|r‡‡‡0‡˜lza=wx|−xvlnay0„nˆ€˜→¤{d1fzŠ‚–}+›8l›!lˆ³:ˆe∞D|15l†–580²‡nf0„nˆ|en2(→ˆ˜‰aa8xu´l©lw„n´ ml©x5a)Dxl©©l†–¿sexŠ=”€x˜r€ˆ–©ly€ˆ‡dozxq–1d|˜l0 0s˜‰–}|uy‡}–(—l‡˜ ae|Hs—%0e|ll 8n–†€7‡V©l)€„X‡)€x„sˆ

1.3 CONTINUIDAD. RESULTADOS LOCALES 23 „ ¤{z › ƒH‡„› ‡ ƒ x l | ƒ 1 I— )ˆ€  y‡ ˜ ‡ yP 5 s™x tl y – l | l s „ ‰w lDu ‡ƒ ƒ ƒ ƒ T x f(x) = 0= ˆy −< 0 −< f(x) −< xx→0  l¥‰l ™l ›2•aw ‡ ‡z | f(0) B‡2xRŠ2—‰¤ €7©l ‰wV‡x lI¦ D)‡„€}{|z › y w l| | l 8›Œ3.— |g0– s ›„¦…f‡ 8(Yxx„› y )ˆ€ ‡ =x ˜ ˜ –}| l  y 5 ˆ =y”l‘aj ¤„2! ˆ —‰Œ €7– V‡ xg— w – ©l—%˜ ‡ | ˜ ˜†x l ƒ }„ ‡ x „ {}z ›2– y7©l x s|‰xl—¨‡ y ƒ„–l Px–˜ †– €  ƒ 5 l| l ›24‡ < δ −< 1 h h→0 (a + h)2 „ w‰lDu ‡rr (a + h)2 − a2 rr = ƒƒ = ƒ $ƒ ƒ ƒ j ƒƒ + δm j ƒƒ h(2a + h) h 2a + h <δ 2a −< δ 2 a + 1m ; › {¤z |T ε > 0 S δ = 1, ƒ εƒ y 8 „¤ w‰l> 0 ‡ƒ ƒ <δ ⇒ rr (a + h)2 − a2 rr < ε. 2 a +1 h  ™• w g| 0– ‡z | f(x) = 1 )l x 0‡| y –}| w  l| 5 ˆ− {0}4.— €7‡Vx¦I)€ l Œ l aP 5 a−s‰{0x”}l–ˆ y  l l¬ƒ „› ‡ x™˜ †– € l  y 8› l| ”y l ¤ ‰w l „ {}z › ˆ1 = 1 Œ – xw —%‡| l 2› ‡ x x § ƒƒ 2µ – | ƒ > ƒ ƒ − δ > a x→a x a x − a < δ −< µ a µ2µ„ ‰w lDu ‡ rrrr 1 − 1 rrrr = ƒƒ < 2δ x a a2 , ƒx −Uƒ ƒ aƒ ax › {}z | y 8„ ¤ ‰w lT ε > 0 S δ = ƒƒ a2ε ‡ƒ ƒ ⇒ rrrr 1 − 1 rrrr , >0 <δ x a < ε. a x−a 22  h• w g| 0– ‡z | f : [0, 1] → [0, 1] ˜  ˜ u %— V‡ € f(x) = 0 x – x = 0 ‡ ± l | y ‡ x¨˜ ‡| x – = {}z ›5.— ˜| „}‡ x —w ˜ l …¦8 „ l vs q1§ ‰wxl „p –†€”x€ l)P/˜ w 0–†g¦ „©l x 0 ‡ | y }– | w  x V‡ ƒ„ 8 › l | ”y fl l S D ‡V| 0 A ¤ q f(x) = 0 lBnŒ [δY¤+1l0(w‰1,a¬l1a)−<]ƒ ˆ¯fεP(Q[x0[x)0,%– ƒ 1,„}<1]„ƒ's8]ε›!l ¦Û | ŸYT8y ‡›„x‡g| §‡P Ta S x→a x8l „ x– P l©[`x0,ƒ 81fx d„(]– ‰˜x−0)l‡€ƒ|Sl<Ÿ ˆ„› w εŒ|‡ lˆyx—‡u w δ‰Û |s|‰=–| y w z › l €7‡¨%— ‡ x – y –†…¦‡ εˆ Œ l  n P¶r y ‡y¤ w‰l x › ¤{z | 1, 1 , 1 , 2 , . . ., 1 , . . . , n−1 s Q –l | 3 n n ƒ2 3ƒ P 0 s− a :s S − {a} >xPTcl©x0‡ƒ–Yxoz iy„}np‡) ‰˜t€oƒ”yilsnl˜ sw |uld88˜Ûga|„“e€‰| s˜›z–d‡˜‡Elxi¨ sl‰|w¤ c|‡8|ew‰oy„¿xD!l–ns˜› •‰—tw‡ w il€|‰n|g|‰©l˜ ‡ u0x‡l l–0i–x‰˜‡|zd†– s|l€yl a| „}©ld|g‡ xYw‡ewxy e|s|‡ )l„{}zx›„lxls|˜0–„›–‡7y x—©l|–0xwly‡ |–}||„ƒ| yyywV‡‡”y}– |€7lx|‰ˆ€w l˜8‡`– |„˜‡V)l €|xDl©w˜d˜s‰˜f| $sl—Iw l)—0— |€ˆ–w‰w˜! |l)‡©l y„}˜x ‡t‡l˜2 l¤a——gw‰awsƒ„ l8|xˆ |2ly„ƒ”yt5‰‡¨l˜ ˆDx©•–¤{zw€ s’ lxg| l˜0¤l”y–„ƒ‰ww0‡z –}|l„0€›„)Dl l©¨}„˜‡ ›x–lx Yx ¦dly”¤|lliw‰|€s7y ll-£z

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1.4 RESULTADOS GLOBALES 29˜ l y7gÛ l |‰€”…¦– ˜) ‡„}‡‰cˆ § ©l x ‡V¦B'… }– ‘‡ ¤ ‰w l f(I) I ˆ[c, d] % w‰l©u ‡ f(I) l©xcl |s w 8}„ ¤ w – l €©  x ‡ w|–}| d[—7yÛ¦§l©0jjj 0…'…gª–l‡xŸYw‰…e,V‡‡†–x|Š ©l+Š5Š—%€→vxŸ„xlw‡¤{z∞Hƒ˜››!+‡xx| l‡f–∞→˜y)y„„›2¤{—‡z–ªˆl„ƒ›+•h…qPq„›w2‡–}‡‰∞(›¨g|•ƒ„‡xˆw%—q0xx–)|gV‡–u‹2•‡€zx0=wl|w|– ||e=‡¥zy%— |0+–†‡ff‡—–(g|+V‡∞|pfz‡xz|[‰sly”)∞0l))l€—sx,xDPxl=+s„ˆ˜¤–(gw–xx‰w©x∞Dl g—Hƒ→x)ul”x„u ¤{lz2}„)‡‡›l−–¤l q©›2|y„¿∞l©S– l˜x0‡[bPal|‡l „0¦llf(,P‡x(gÛby7lRfx|l)D](5‡Vx)yˆ–all=}–€ |s€|)lè w ›¨”yb−>−„l h∞>a˜ ql §sa„ „ƒ„qy'™sf‰w (zly•lD7€bw)u„› )„€g|‡ −–}<0¤‰| –S gw‡‡bzal| Is ˜ PxPφl –™((b!›5|‰x)–) a ‰| – b fx˜(‡ x‡ t| )˜ˆ — w |y‡x §=8‡V„€€x¤ u sP(x) y −wg€ˆl l >0 ˆ Œ lP(© a—g)}„ –¤©< 0l „ w  l| ”y l[0”€ …, +8}„ ∞‡ s )§hˆ w ‹| y ”y‡Vle| €7…l©8 xD}„ s™‡ %—‰| ‡V‡™€ 8Dj }–‡ –†y†– Š 5s l „s |2}– | |2†– | ˜‡ § sHl 5„ –†| y”l €7…8 „}‡A–}›! ul | l©x f(0) = 0 0Eyw0)‡|Alq›| ty–}g¦|–}e‹|–”yolz„wl |¢r€””y e…l m‡d8l©l „}€x€”‡¢la€ˆ!›‡dy˜l€‚e”€‡ ‡r˜ˆ€ §WlR†–˜ | Ù8‡e”y D‡li‡§e„}€”)Úy…r8d8s˜‰|0}„t‰‡‡‰‡‡rˆyˆa˜ s–}˜ ‹s ‡‘l „ ¤—%x ‰w–V‡ u l€ w „ƒw–v l I| |–}¨›!y”lE• wu7y |gll |‡V0–€ ‡˜zl Q|!›lHw0‰ |‡q|)–}ۉy| –}7€7y| ›!l w €”…¦Q ) ©l¤}„ ‡cx„w‰%—lj ‡‡Vyƒ„€A7€ w4‡ –}e| !›–}€| 7y• ulw ”€l|g¦…| 0) –}„‡˜‡Bz |lŠ(fT—%yy7©ll˜0)d8¨SjjjjMIl–†–ƒ–†š)|‡xY¦|0xlV‡†–†–eŠ‰|nƒ‚py‰€7„$‡|ŠŠ“x18u€`ozIo„›‰‡I−)z©lyw‹¤†„,0áӐnIr‡1Yx€€xŒ©lw‰}–˜´8Šˆ1dy‰e0l©˜u|efd‡˜Yx2Dn–sllh08hs˜(zm€8syqh‡Òc‡x}„M!§)w›±0}–Pφƒ„©d)ly|)l|)€adr—D¦0)P¿„(”x|l7y—›[­‚w.zDw‰‡‰|x”xa˜wl−|l©B²)–Ig|wyl©d´)8ڀ”l)‡w‡xy”,ŒI—·zn—xv¦…©€0q„›l©l|‚be|Hf–<)˜˜‡D8çxx(l]T€xfI—‡V‡jy–†}„‰‡swx„8yw…¦‡ny:|K“xs)—ˆ)—B„5†–)xx8˜[y€ˆŠ©lˆw‰˜Xj›wla›δ„}–©=TlvxWl©|7€‡©lŒ‰—‹(,¦}„lxx–†lzvxcw–ll‰˜w€b‡V€”|”x¨u|j„|)0l€ˆ¤%—l7€wx]ƒ–Py–}–y7Q›„Š›l0}{–„›w‰z‡€‡→lw|Tl˜—–†|˜fg|[l|§€‡|el|Ûg‡ax‡0˜l‡suu|f©l‡„›Dx€75,‡˜l)s§¨8S[(y7xY‡|sbP‰—axlxjl”y„›B‡Kw¤8])l©x„w7€,1ˆSIcŠ‚w‰|eˆDwV‡x„‰[‡b2'…D)ˆnaP‡l¦Ix›P‹‡]–BsD−<›Œy,0ls|x‡5¤lD|Sblˆ€l[–˜Ÿ{}|yf„z˜w‰ayyhx‰|]|‰w‰w)lV(‡8y‡ys‡l,s2‡}–xxlD|ˆ–}8„d|g›„bx$|xƒ„˜uf)y)g—„0Œt¤Š]Pw§u‡„‡™l—g‡u‡−<€‰|©l¤lcwgh§˜syl}„xc•w‰y‡©l–†l[¦…–}wssf0Vul›¨8a‡)xYlx8(˜–lg|ll©0B7y„)yxƒ„,™lx„†)lƒ|‡|Yxl€ˆD|B‡‡Vbδ2fy‰xu¨„›yl“x–”y€Šd˜((—7y]‡l)l‡z€ˆlz¤xcl˜‡|‡|…|l™g||ydTgw)‡V)‡`|l)l©˜0xfI€lƒ˜jxÚ2jéx)lR„‡n€l−<txD—˜“lnxl©„›{ƒy¨›Pz{z—%¤„wlxƒvI→K„}{¤›„zfww‰V‡–‡n›|[˜ yf0Ša¦…|e∞€ly–l0seT(s‡l)y),‡€ˆxs‰˜l|xlYxf„}lMbHxƒ)‡V™„Iy„“©l„(˜yx|]w‰n)€Px–}wxD‰˜MD<ˆ)|I©l€lDs2I‡lƒ—I1{ƒw=x[zu0)©lI„|=a¡ˆM)ˆ‡axŸ}–%—=,€=–y|w,0x‡bylf‡)zy7|w−a€7||ˆ[]x›qll©ayB‡M—˜+ˆ–2€”Œy‡xcjg¦,(‡K¦…b1†–yllJ©lbs¦…–8)Jk„e)l˜jzXj‡f]„†€T ˆ|s‡–ƒx—‰=ˆ8‡{ƒxzŠxl…2)s'€7„›Š‹slug—'‡$¥¡f)lP—‰|8φ|S‡a„}x“jlD¤–€7Kyx+)l[2©([8I'‡‡a˜‡xaxl©–8bn›„yq‡l|guV,>Yx)|,−,l©0bwy‡b˜‰|˜b˜†–1©l©=])€Dx]0‡‡mlk€xŠ„sss

30 FUNCIONES CONTINUAS1.5 MONOTON´IA. FUNCIO´ N INVERSAgfDx—%l©l•l©l©„ƒ¤xjj uwfd‡|xxxsgw‡Vj(<eg|}„e[f€l••}–xq‡fS0fiw−0|8cz−),yg–xsD§¦—|ež‘’nr1‡+1z>–†−„ƒlew0:V‡‡V|i„l=‡—|IxW:c∞–1c|€€0fl7yV‡˜qyfz(i•(i(V‡lyg–†|yow(l−elxly)S…¦‡›„€•yŸŸ:–|gnmn)w˜B)sB∞—‰lllx)³‡[0g|teˆ¿lx›h|=xh›ql€7=→0‚Š–vxe–l›„s0‡V´,ˆ‡l,áz—gg—y–Š−I¦y[+|!˜‡‹‡–)l·0fz„}S}„)l)‡e|‡–¤1(xx,∞|tq›–†|s)€xxwB|n)s+sdlˆˆlf%—)g¦…8„ƒ)¤€„¤n∞XjgshcV‡|`lw–−S<lfuw l˜€€rz→0(˜8(ˆ€•B{E|)zx–}exfl–w1„}8‡h|flbˆ¤¤(gÛ)cwŠ,|eD−…|¤yw‰wf‰|i+¤–5)l80=l17y‰weW)fl–x“–|–€l)ll∞xln˜j‡ zxw˜fx€ˆ)lj˜fs|ltf(8|eYx¬‡()|e–†(xy„y|“lf˜xw˜7€˜)T→|)(¦§7yR5j¤–)Elm•8lxdxlslw |−>QywV‡)=,ƒ„}„eg|%—8˜‡`€5yy„ƒ|©xc=f0q‡V›l†–‡x{¿xz¦…(–r€•¨›x‡Py•wlzˆ©e¤˜fxww||)g|gs‰wc2¦dg|l©SˆŠxy7i(˜f0ql¤ xR‡0lfˆe0˜xl)–:wg4‡‡–xDynwRxl¨)zál©‡0S©l8|sz||ItYx€%—|˜l©„ŸfÙy=lel)→hxYx‡Vwf7€0xy‡qlŠxx1€¤–|:–€7•™l„}˜lxewy¤)ږ¤5˜Byf‰||l‡2n8|g˜‰w8–¤−y1ÛgY7y‡|‰›©l¤–0l8Sl™(1‰|ll©Sx–‡›0l©(‡llx–xzjieIxx)˜||˜n¥‰lms)gwqy7<›„|→llyt5©ll−)ˆ„›7yo=r„ƒeYx‡€|l1yi0ync|¤l‡5c˜7€€:tˆf0o´‰w‡x–¤tlx1z˜‡xui€[–ytla00lvl„˜lˆ—%y‡nT–o|,m0a)l|g‡e|ycx+ny7–›|˜–l‡˜lxi,elay”∞|¨o´–¢Š–yl|nl2x˜ly”Tn| llle%—t|ql)‘‡–†xP7yj|B|enV‡˜l€i,flB…‰—→n€©lSl(fy|cl2›S1˜€7xY(0vf€r‡VyS(€)wˆP[e(x‡e80¦I0l5)xr|„<c0),Ss)l©l)s–ˆi‡+€™0zx˜lae˜yy=‡f|∞l8n‡¤¤(2›y7d„we|˜ywgw‰tl)lƒ„)x1|ee‡ll)xŠQfs1[Py7%—l©¤˜l¨S‰ë αl(x|¢gwf.l©V‡−<e|‡2rap7|—,(€l—o`…o”ƒaβ}„0l)f˜‡d${‡p€fl+(k5]ê>´8†–x(s‰|„‘xkŒì€inD){}α}–zlIeyls)„l‡{†|eÒz|exŒ{}‰|ˆ€z=|‘d)„0±%—›„−<8ƒ„→{}˜cz–Vr‰ÛsqIj›a©}„˜‡l——%7­›–Is2›—af•ˆ€dy7|›„'²w¤–wwV‡(ê‰wD©lε©´f•leyh‡ξ·|€g||0‰wxcq(–©l‡|fsŠ>)xy–0xxlg|ˆly”‡s!›ll‡z–)<ˆ€ˆŸl‡t|||„fzvxŒ0l€s‰ws“hl)|yl›hws§‘s˜l©xDlDwV‡xw0%—›!fxuV‡l7€H–g—|+–δE|l©ll‡:|‰|‡ }„8„ˆƒ„xI||‡=I‡tlε‡2€–}|‰s|Sxy”||usl→slll˜ξw˜7y–}yDξ8zfˆ€€„›hs‰ll–‡V–›„›–‰w(xlP−”€”€x©l€7fβ5DlDg—|¦…€ˆ0‰|lx‡xV‡u)‡(„}ˆ)y”7€alDx‡–¤›„|‡a¤l˜€|‡„}©ssP‡4ly‰—w‰Š‚,y‡>)—%‡–}0sˆ¤–}β€l€f|||S–‡dw‰w‰ƒ{˜(z+l0]wltw—%‡€xŠzlll|€”Q–ysy–‡)—g¦I€ˆ˜l7yε˜f§8‡xaS—%Ÿ˜l}„q–„Pe|}–lδ˜‰|V‡ˆ€ˆ)˜l©˜‰„›0¤¤ƒ{z€R‡‡„)xY()l>)w‰w‰‹y‰s|<lg—x|lƒ„§—`,llz|qf„˜|}„0‰Ûs‰w‡xx‡fdSd0‰—l‰|l©ls+a(y‡00fe„−€7€˜–ξ¤8j‡‡(|yV‡†–tl™ε)„$aw‰•yf|yŸg—aw‰)(wxRlƒv−¤y”<x–aˆl˜˜ll©|l<‰w)˜fŠˆ€€7+˜‘‡€–‡ysldsl…=‡l©)δ0x+8†–[fx4˜‰—lg|als‡=„}S(x…‡u7€ˆy•a,εw˜l}–€–ul)b=−›ˆ)y)—7€sllx])g¦}––}0∞>V‡sl|‰=g¦0–a–f0€€7ll‡}„•„(f‡›ll|€f|x}{§<z(y€7(“s7y‰|xxl©l)w‡†–/zlzl‡V•xx|):)‰|‡Bs(x”€—lI€y7:αS…¦¤–§¨€Tl‘lf0aI˜ƒ{d)<z(0z`‡‘−<xˆ0lbj– l<l–5%—P¤ξ−%—‡x¥$Szˆd‡‰w|‡V)lx(<€xaQ§luj€‡)a=z©l˜ l2›}„−<,P¥axlx‡lŸ aŸ2l+l„‡β!›lI—§›hxx+h›∞ˆyˆ‰w––—gu¤¤lŒ)l§—eδˆ‰||‰„}|‰ww‰˜l‡V„†‡)‡‡‡llll€ s s

1.5 MONOTON´IA. FUNCIO´ N INVERSA 3123yεf…—‰—l©l$¥}„l–}›l©˜0¨¨SŒSj 2‡|†–−||‡vxxYlŠfx€7..–w‰ll–†‚pp‚y7ylx——=s7€|‡V−Š1,8¤©lwo`o`l€70|[–†l—g0(yx1x›2xgÛ¤–€”wg‡|4Œy”„¿b–‡yfk–Œ…¦(8´8´,,|©˜llfw©ly(Œ„›‡yŒ)–†z)DnnDwy}–yl|˜)x—%|lx–ÒÒx„})l–}0%—P]€0=›‡˜)y”‡±00±P–|f7y‡ˆ€s'€©f„ƒuVl‡rddrI‡˜f(luw|luh§f0€lf€7:wgfa|­‚‚­I(€”a0x”–‡u|…(sB²B²f…¦εIˆl˜I†–˜u)l©–lI2›)´´)|yˆ88y7›„(l%—€)··Š‚sx—l))−8qql→a}„>›„l|›ˆh}„‡V›2‡ˆ‡jS˜`„—gx‡ç0Dˆy7−D©lŒ€δ„›|‡w‡l–j„}lV‡0x‡fjl–<}–)j5xD‡|‰|xˆ)z<(¤|‡x,‡|–†eslay¤‡VIhyy8PŠfw‰|x”y‡€a)–}§w‰¤8€fl©Df(Œlyl|lv‡|e–}ˆz©l„¿(ax–‰w!l”€f›hlˆ¤yw‡…xx–zji¦…x−V‡<l+¤‡Œw‰0|§xll¤,8—g€–1w‰w|e‡ll)€7€lyfw‰Db„}f<)x|–†¥|)Rl|€¤–z–)Ûeylua(l—©‡|l©l©ydz8agÛ[䁩yld=l–}˜xDl|y−j0)y”ax™›z|8)d„|‰|0„¨Blwƒ––}„‰w=ˆ„wu–q‡+<x–˜|}–yg¦−D‡f€jy|lDzIw‰u€lslD¤–(l|„ƒuly7s%δ›l8lzq„a„›–zεy0w‰l‡l›„‡¥5|›‰˜sh§‰z8–,¥)”€xl¤{¤z}–l§A|‡lg„la|–…¦šwÚ©–}‡dl‰w)l|u|–x|jŠ¤¤P)yf0|—gxY†–8”yl‰|+s§hy˜„}†–ww‰y(ƒ–fl=ly7“„€7W‡‡Šc‡„l¤€7xulf(f„ul–lf–†ˆ=g|εW¤–l©›‰wa)(„ƒjIz¤„©y˜f€]w¦!xI[—lŒyˆ<–ªaŠ—I(˜l–+)©lul|”x¦…8f–Ix€ˆ•)lx‹€s)hw˜yx(›−w)wffx€εa8Iê€ly|˜I|g(y|f>)„˜ll>€l©)„}εy‰|zl0‡2|l|x‡‡Pˆj−¤,¦)‡l˜–„—¦I7yaf|ga‡s%{†‰w€7z„›zl‡¥<ll2›(lˆ€f|l)jfŠl‹x›„„¿a+„f0Yx(x©lsw–ˆj–(y|–xfaf˜–}y”xD}–‡−–†€—%l0a¤„›„|2(h|0ε(RŠylsl‡)x‡y‡w‡V‡7y]…—%δ,D−fes‡B‰—„›m0lƒ„|||e)f˜l–‰—)‡V‡)€”Šˆl€7¥ˆl)(€yuQε©l€7‡“€7y¦…ˆ|‡V|a–}xx=l©8‡V“x)—lt|)¦I¤„y”x§Dx,l)©l›„ˆ€¦IR0l}„w¦l–wg—sf„x8f‡˜‡−‚€)fs‰¤|‘(˜‡g—(„„}y7‡l−|€!yfa¤–szxds‰wIx‡7€l€ˆ©(yf—l1–})x%s‡VV‡Slx)¤s+8|x†–(¤ˆ|lPg—/€€δla|y‰wy”)¤©l<‰w)lw–ε|–l˜xlgwl)ll©”€l!>!›>−”€)yf‡|€axYa˜l…mV‡0(yDlyjl„l8I‡fεe|−0—%‡¤{→ufjzIw)|˜›(„}|)x˜‰Û„ƒs2‡V‡as|Eδb)l©yl©1l„e|€c¥If¤¦…–}€–x−xs0—%f|(8gw—>8ê⇐l‡šÙ)−)I)V‡w©lw„}„}fŠŸlw)|¦›|sd‡‚€Yx1‡jl¢|⇒0s|y7yI—¤ƒ–xy”€—%(—›„„†lŠl7€wl„Ú8yQyl©‰w)l%—‡Vw‰‰w|–¤‰|‡€‚)x–€g—|ˆ€f)xflf–€‡')l–†Dl”y|‰–˜–y(‡V}„˜l˜u˜˜0‡B)l)I‰‡‡=„¤l)€xy‡‡‡lfll)xŠ£–„ˆsˆf(a − ε) < y< ff(−a1)j +ε = a − ε < f−1(y) < f−1 j f(a + ε)m =⇒ f(a − ε)m = a + ε.2p+E1ygÛl0•© w–†¨..‰|a|∞j——|gx–e–}rwu‡D0 msls––sS‡‡e|fzz Æºllpli|4|0|„f)l|ñU3g–ƒlny€xo–}–}Vu#‡Ç|h|í5 sÓl©lg|˜‰%…y7e‡xYslllÇhnsy›”€!$—€©l7€ ¦…î€7xDx–}iw‰#0l–z8m()suyíl©)(y„†f€7¾xt‡d‡f8¾p−¤–Çð−€›©ï—Ia–}1ÇÈ1!›˜#0)3l(r(ˆ€xhl|òxs˜¹u)”y)2)ulfÙl ©=l'$=‡l)|A)„ƒˆ¾}„5x˜€ˆ€)Ú3—xn·l—%n·)l©†¾0g|‡VYxx•nxa–yw€™ól‡ €7©lg||©lsP‘)h¤–ÛfYx7y0xY‡YŸ‰yy—l‰y}–r‡˜ôE„†‡)z…'z3™‰sll|›s –˜“„|˜˜¾˜©l¤„llhl!x–}lۉ[||¨ËÛg˜0›„•‰|7y7yUñx‰|,‡wlll–‡#+–e|”€˜|˜εx5¦…0%0∞ x€8>l©Çh– lV‡!$s§‘„†xz`§)‡0|¨‡#0Bs0–l©l)l‡[()§sfx0|x x(¾¥x,7yD0xlx0‡V+lu‡V‡%)‡€sˆ€ |›„∞|ÇlB=δ'Uy”y|yˆ‡u–}%–})=| |x5xñfw wn©l(u€sôεsx0U3ˆDl©l•§)l[wlyU'x0Œ|=V|g|f)(,–0ˆâ(+05¾‡[50ƒ–0h¹|‡∞ˆz|‰),§y|ƒRt+‡}–=<)|xwx∞ˆ wx→l„|gδ5{¤|Bzt5›)¤+sy ˆˆ ˆx–w‰©x∞x SH{z‹li• ©l)if¤|x„ƒ€(‰wnƒ{xz0s)ll )‡˜Yx e|y”„ƒl„¤=sl££rrrrrrrrrrrrrrrrrrrrrrrrx l | x l | 0‡ x x l |ƒ (a + h) − ƒ = 2 ‚õ a + h õ h −< 2 h = ƒƒ < ε. a 2ö 2ö 2 h

32 FUNCIONES CONTINUASx+gÛl©l›0˜ˆ–}xˆ |l€‡€|x£l|‰∞7yyyll|lxۉ†–€7l|00−l€™s|‰¤–”€j––„d7ys1ál−l…¦€–„ƒ—ly}–˜(||©l!„ƒ8„!|)¢xπx27yy7l©†„x™l©7y›—h‡)•l2—lxY••mlx„w•§`wy7wluw€”lI—=§llg||g|g=€…¦l©|g|)l|)y0©DYx80}–zy70x€‡0ˆ€–|Ûey)––}„l‡‡−–l‡‡z‡€77y‚€zl©z©‡||s˜z„f¤–|s|ldx01|‡˜y[l”€y}–€−x™0y}–l–}¦…|7€07yl|8|h‡§Qƒ„Ûg–}–¤‡u)”y1˜0™h|›w…x l‰|}„x–,2lyx…x‡„lxl€”€1†–lx8ll€|…v€l|w]|›=I—lx€l©8y”ˆ=„¤y”||Ixl4©xYy„}qllj‡Šy‰0€¨ |π÷á2úülQxl•zCøûˆ7yy7„w[ylm•|ù0÷0€luw|˜‰˜g|}–‡xxl,−|==|gll„›Dlˆ0πjy”Ûg‰Û10–v„€–x©ll]‡lx‡t(–lz‰|‰|1x”€‡–}‘|x|+zDˆj|†––…s|€v0)7y–˜x07y8l–†l4‡l„„ƒ‡πB|l2„ƒ}„|‘§A=x|™‡€”…”yym(l…¦§syl‡•l0|˜•(–}8w€l©)w!›)|−l!)l§„†x|gx‚€g|‡wlx™V=∞!0R„x00DŠ–y7w©l–‡‡j‡Q‰—–†‡,uz1−Yxzl|||‰|0+€‘|y}–|xy”yy‡h|l‚€§π2l∞}––}}–0}–„›–}…||€75|h,l)h|–lD…wwyxYx……‡)π2€‡88‰y8−lˆlxxm››}„”yz€€(‡ullxxπlljX˜||xx|k|j)¡l‡−‘l−y”y7l˜x5Ûgπ2|=`l2lld„l|‰πy2π2s:| –}‡x–)l˜‰+„ƒ|−,k−¢§˜†–!›™xYl›”yπ2m1”y1ˆll©l•4l(kl0€=”€sw|zcxx§rly…–†g|7y„¤—g)‚€jD©l|8¨l−x0–}€l)–Yx}„y”7€)l=‘l–l‡yhx•l∞‡„‡|z0€7w”€|xDy[–¤–y”)l…e|–})−x‡$sl!›hxY!§88y‚€0x¨›yl1}„8l›–€7§I—‡4|‡y7,›|‰z–¤tx)u1s|ll‡B0I—€yll|]|—‰‡jŠ8ˆ„“|)†–y”πQ2|€B|x›ˆ€l4y”—gsly0…−2l 0–†ƒ„l‡s§ll)l©|8m–|˜›„˜€Yxxx‰|wxy‰”yy”l©©=ll ‡‡llx£££z1.6 CONTINUIDAD UNIFORMED2δT…l˜Œ |δll¤–εe˜|s ¥yfi>ly‰|‡Vfh‡)—nƒ‡7€:|I0l—|‰!iwD|c˜ ‡4wS8i•˜lx||o´wδ—→l–†)y”€˜n|g€„™l‡>…‡c0|z—ls‰5– ˜„ƒ‡0w˜‰˜z5—%ˆ|l—|lly©•Œy)˜wˆ€f8ε‡l™„'clg|„ 7yl©¤ §‰0˜0sjwx‰ws–‡¤–ˆˆ€‡,lz˜|0|lfŒ‡l7yT (ll2|l©¤xss|εyx$)‰w,˜‰Š–}lmyw|l©l§€“Isy˜w‰| fƒ„8P˜8w–} „“†•|gl©ld‡Su|l¤xl€‚€B¥|yyw‰›u)¤€‚z8wl‡Ûgw‰nS„l€ˆl©©›lƒ„i˜xfs oqll2¤w ll|uδr‰w8‘||w\"€7ymˆ|l |I©lzy—Iƒ˜eÛ¨e‡V0x)‡©€7m‡€'&s4−‰||y ©§T–¤™‡ey ©y}–nwε)|5 ƒt€„¤w <)>e”y $wl)€ xs¢ c8δl0ƒ– ¤„of|)¤x€ |en§fll©lwStx–y¦sTily– n0˜˜l€“8s‡||ul—|˜˜luPawy„ƒ‡w‘–†ƒ|f|lS„ƒεy(©|•ws‡‰x€w>)ˆsSX’|g−lŒ0δ0=Wl|–ws‡|fz>ƒ„|SS(8y5 yˆ€g|ˆ0I„›f) ˜ƒ'¥ –jXl©D<xw©lYx ¨›xYy‰ˆ€ly”εxz|l– ˆ s s3x2aE1§I—x —l1...)δj———I—ˆ€ey=+)–mlˆ€¿¿ y|䝌jp‡w‰‰w ¬l2δlRRq§˜yl)l)l mo‡˜•xxƒ‡x•f2wsw˜–¤(2xxe|M<‡x|g–ll =0)0 x–x”¤1–−‡V>w‡zw‰zε2δs| |P%— lf–}0=‰˜›h‡x(fBx1|yl2Û —g:δl1)Ÿ(‰|‰|D„}(„›‡0ƒêŤ– a‡‡©<,ˆb‡Š8T)→1©ll)x|s%δ)λxx€˜sl¤w5wƒy |‰–}•ƒV‡ 7€ › l › l | ”y l DV‡ | y }– | w  l | ˆ[0,∞) j Œ {'z †„ ‡ )l xl | [0, M] |E¤‡ ‰w8|‰‰w ll©H„a—I}–l x•†δ)¤‡VS…l€‰w7€i<lδy›ƒpl7€–}†–s>1lÛe0wc› ©¤„hI— <)08 l)i€—| t|€ˆy δzI—7y8ƒ xli„H) <ay1ˆ€D‡¤n−‡˜‰w1a|x‡l xyxl }–a2=l|rr| xƒ w>—2D=aw‰−0+l©xl2δ–s˜|a<lS)2δ2(Q|¦ λrr0δxx ,<luDw>1§‡€ ”y)|˜10lrrrˆ x‡‰x| 1–yI—1ˆ˜˜8)l€„g−€ˆƒ{ˆ€z¤) gwxq€1y2rrlx‡†„ rrr‡2T˜ =xx‡−,— aa2>rr 0 x−y . = w | y‡ x ˆ1 > 1 δ yP D

1.6 CONTINUIDAD UNIFORME 33 —„}› †– —dl l 7€|‡¥ y”Œ ll –„©Ú 0f¤–l‡8 |)l|e| wxy |g–}l| i0l pw–ƒ| s˜[c0l‡qh,| i1€ tl[]z0s i,{†za‰—1w n7€]g|¦‡ aj¤D) ‡‰wlg— |l}„©l –}x)DxDd{)z•i|8Bs„}‡„˜l x |‡‰‡©l y€xˆ ‡l l„|g|V‡y7 ll©€ y‡Vx ‡l€ fŸ˜l l ¨›‡)l›„“x [ —gaw „}2,|‰‡ 1–}s˜]•ƒƒ„‡Vl™07€ ‡ ›l©•|wYx ly |g› 0–lx ‡zl| | ”y l 0‡ | l©syx“†–— |w lw|‰€7!‡}– •†V‡l ‰| €7| ‡› Dl)l x£s – ‡z xI| Š y a> · D 00ˆ – FcjD¢ …l¡¦ualäz nu x§Acs‹ Äliuuq„o”x ýrxn0›!aˆb„ƒŠe¹ st¼wg§©loxd‡ex2) pw¢€x™ó ¤ˆ€op'w‰–}l|o´oÇl„˜(%y7gn“llþl©ie€„sIxc©lnzy0¶ax¢‡cx |i˜˜ aÿŸl•l wwl„}² e±7|e|„ƒ0nDsy0‡Ò ‡}–y|V‡˜«ŸS| lwlB´ l©oÛg|l©V®x g|‰Dxy±7‡sa¤–0n 0cp—%˜r–˜´‡ı›2d‡z²}‡tfq|€ m|„– g×7y‰˜‰˜¶vl˜nill|²¤cq%l‰ÛDn axr©s‰|0l•‡s¬ ¨'w¤– |0y”g| – l‡w0z ||–‰| ‡xzs}–†–Œ|ƒ•‡lV‡g— |„7€w€7l)›0–†|‰|x‡l†–uV|†•D›V‡lŸ‡Vw”€7€ l|£›||$¥y y–}”yll|‡l›7€w ƒ„0l ‡ |u‰x‰| 7ys wylz ©¢}–|‰|Vã 0–¤w㩁‡‰| xxsy ˆ –}— ¨| zw u¤„  ˆ ˜ S = {x P 5 : Bδ(x) ¯ S =W  X Tδ > 0} . l ›„‡ xux ‡ƒ„ 8› l | 7y lsfx }– | ˜ ) €h„¤ lV‡ 2›€ ‡l ŸxYly ˆ€h› d—g0 –}„‡‡z |s s (la„ , xb– )u w =– l |[a”y l, b€ ])l x§ w „ ±y ˜ =‰‡ ê 5 ˆ‹ | w |g0 –ƒ© €ºb n G ¡ g m £G ¥¢ ¤ PfT•˜±˜0jjj wƒ–(†––†‡l‡–†}–eŠrpg|–†|ŠE›2xooƒ„Š‹0ys+q„rp‡––}„‹t5{}xvz‡|{}→zezYx›„i›•qx€|wyxDemwl©0€ˆ{)–zYxdw|gy”sfaya=©la|‰0˜7€l:˜qcx––¤d}–|w‡S‡1•†zf˜V‡ey=˜|.(S8→„ƒ7€8sp‡››ˆŒ¢|.l01)–w¨ 0llQl5Œž)|f—%f¨›|fld„x ‡–(7y–l)yx€|lql|8“x”yla|wa)llz„¿…07yw›|g|g€©ll>T¤>z|‰0lDxw‰lp}–Dƒ–0–¤Š‚€7•†)1l0,V‡–ˆ‡V‡l„qs|€7f|ۘ›yaqf(ŸYy”P–}l(s→‰‡l„|„lq)}{>zˆw±››¤„−)jXQ=˜∞ˆl0=lq|lˆ€fˆa¦Û|y7(€• l}{qYŸszwalڇy)0g|0q‡=–‡n0Tlj˜s –ƒ„||s­‡‡0z¤yy”l)|z–}‰wPlxbx|§wŠ†–lfwa›¨Sd¦(qlsqD|→e„wj)V‡l}{zfŠ›h›+|¦ó±|˜=s%l)Ÿ∞lE—gy7wSx5xal„|–'sayl|wq‡ylaql|e2‡˜|„¤›2…>y‡r=dl ‡‡l0€70e|g1x|–†¨B‡+cgہwyy7uz˜j)∞l)l›xlhxHa|˜–gÛêclx‡Dla‰|j–€7i¨ ‡xo´‡‰z˜–––<|˜axYnl ”ya—%±—l˜1‡ fjl‚Š<wxuˆ§ˆ – e|7•ny 1Š–ªch… ‰¥sd§iB‡wll©oz Š|‰›2xYfny–¤la©‡‡l©|xxxla0 = 1¦ n ©§ ¨© ¨ ©  aan = 6&&6 6 aa−n= 1 anam/n = n am b1§lw¨j †–I„w12|i†–p‚„ƒ/nŠ¦g„ 2`o}{nz|o–2›Œ lwk€ˆ´8mzl=€–›Dny7y {†z҇liژ10±loard‡7€%—ˆln—‡­‚>„dV‡tRε²B£ “€´)©€ˆe©lz·¥1>dqx„ƒ‡V–}0xˆN—ˆ!›‡ˆ€–}0¥$‡V–}jesB©Ú|Ia–ƒlswŠ¤x”)8a©llw<Œ„„›t–xs–ol%—−l ‡1©lnn€‡rxYx‚Š˜xyaêP>–˜¤€‚y }–‰w)>ª–rl©…h1|lyfx ‡81y}–qysB››h8„→)§˜}{z„Iq››lg— –0¤r|u„}+¦g,¤–w‰”y8 ©–sla2›llz |`qPˆ‡aj€x›n−–=±lus10ww –1−y„l8<y8|d„jX–ªr„e—˜y7©lεlh=†„lcx Q–¤1©ll©¤w s¦8m|nxgw|e§|y l—™˜[0D0r2‡•†‡0‡,V‡ |‡¨|+<—%€7| !›m∞V‡„}˜s‡€ ¤–ˆ“x,)0ansx–˜ ‡r–†za‡›2|“ŠP xfm˜–†¤§r„ƒy”/}–gw)l•nvˆ„¤€ll| € Hƒlx”=alfl„› e|‡o´sg—›2(0r‡ >aƒ–m€xf‡w‰myaulsa))I¦rl›qm1−aˆ/¦gnr>¤d– w‰lezl©>|1lxl (1 + ε)n > 1 + nε > a, sh§ a„ w‰1Dl /u nh‡ −0‡1›„<‡u儃–ˆ ˆ€  {z n­ ©lz x –†›¨ l©Hx l©Yx y 7€ ¤–  y )› l| y7l ˆ € l D – l | ”y ¦l 's xl5y – l | l 1 + ε > a1/n

34 FUNCIONES CONTINUAS Œl l a| M1y ‡/n|g> −l)0x 1‰s δx <l= εnn1ˆ Q x– q Ó± § 0 < q < δs‰xlcy – l |l 1 < aq < sa1/n „ ‰w lDu ‡ Mˆ § ‰s xlaq −l k 1 <˜ ‡ P r y P 1 + n(a − 1) > Œ – qP ± q>ny –l |  8„ ¤ w‰ll©w„j ‰w–}x|‰–†Dl†– }–0Šu†• V‡‡<ákŒ7€ q›w  hl→%—˜„ l|{}z‡‡›<›+y |‰‡∞εl–δ|gl| aˆ>|7y l©ql˜‹ x€a0‡4|0=qls‡y %—„d‡0|>+V‡„‡g|y†{z€›„∞›2}–a| l©lnw‡–xRˆ”yŸ ll =qx ›h–„l→q{}z|p›g—→j„01{¤,z}„+(›−‡fq+−a∞¤∞Pq(‰waa±l¨,=q−Ms ©l l©1px1xv]s)¯a>˜mS nlδ± >q„ƒ>>q I—§ 1)•†10s‡Vpˆ€ +7€qg— y−!› €78)n‡V„“q¦(˜¤ al∞=1w‰ ›„−l Msh‡ „1rrqx‰wa<)>lDhw >u δ|Q−0‡ s‰M%—x)l¦Û1—lx'‡YŸ ,rr‡‰0y0<ˆ¢‡– ˆl |› z x ê aq l©x ‡ ε  w ) | ˜ laM rr aq+h − aq rr = aq rr ah − 1 rr −< aM rr ah − 1 rr < ,¤ wgl©˜ ‡)˜zw ‘||‡g| ˜ 00§ ‡u–‡ ‡z| „ƒ|s yy ‡–}a|‰—˜ qw7€ V‡Vsx —gws „ƒ–ag|l©–†xx˜•†V‡‰—87€˜E7€„g› V‡}– j|g—l†– –}y”›– –†©ll Š ˜”€l …¦`§| 87y˜ l„}l)l‡„ x 0 [‡−—‰l| €7‡dMyV‡ €}– I¦|l ,›!wM˜  ]l xˆ¦Û0l '¨ŸY‡y”ls | R‘8 |x 1| ˆ ‹‰s [x−| cl y M‡y |g– l, |l ¤ ‰w ±l ¦l ¦ ssBl I—y € –)lz ˆ€| xq€ll €©w s I| —I˜  ) €ˆjBM— zw |g© }–> )˜ V 0Š ƒ„ Rx – • Ml©x©]hs ¯ ˜x P [−M, M] ax = „ ¤{z › aqn.   qn→x qn [−M,M]egd›2|–}}–0l¤˜ |h|'¨xwgwale…7y‡4”yl—rpllI¦lbi€”„†€o|wtl‡‹‹¦…Hƒxam|nxe|xu8|xYs‡†–l e†„‡ysxe|(‡oEn|a0}„l˜‡ a‡4€„l),(cl©nl•u0∞xD¤xYiwۉ©aea}„sy,¤w‰|g||g‡p+(…¦l)w‰lux0ex–}xˆe‡—lD∞–”y)sl)‡ˆr–lz–l©‡xq˜|zi|jj)x‹a|l—‰¤lsljonl©7€|wg0w›„„w‰g%¦‡V„x‡z‰wol©|‰yag—‡|l˜Dl}–s}–y„rqx•†x–y„u)8V‡©l¤l†–i‡y€˜|™›˜t‰˜7€w‰ˆ€s›mlew˜lqlg¦ %—‰Ûs˜lyl©l–o˜8‰ÛV‡‰|lYx›„ƒz4l4l„›|y—€‰|–|e˜‰|l–˜}„–n•˜x|‡l)4‡l5wl yy7xY™s|•g|bylq©ˆw¤„ƒyl7€l0€a‡w‰|g|r–¤y|–l s‡2lx08z„›y˜l©e–}|ry©™› 8‡uVxl‡‡ ›l|ag¦w˜˜–}x †„h|l©–|Š‡‡l…'l†–x€…zBs|˜x|}–I|5l‡[)l©l”y˜−Š5€„xHlx{}zx–lwx„›MxDl©lf|y›2e|V‡YxD–}– y|˜‰7y,|0ƒ„‡€7©lMyly–|¤–wxq}–‡z)z|‡|‰xI|„ƒ›yz]˜wly$s8–¤Š ‡©lDhl›w0lx0|e|ˆ lf¨e|–ly7fu‡z%—|`qlu—sHQ|ny7‡‡V‰wnl‚§•lc|—€w8©l|cil„›|˜g|o¤„xi|gl)o´y7vlw0n‡l0Yxn– y”wy|€7‡e–ƒz€l)‡–}||‰‡slV‡ez|„xy0¤–€xl0‡pel0‡V)s€–p¨|ex€”—˜oƒ„7€l%—o!wt–!|©l‡Vn˜e˜f|§qx•|‡nwey”˜‰l0©ll‰wng|c|g‡©lx„lli0cx2|0a|–l0i„¤‡ƒ–yolayz‡V8l–ª–|-x££l„„ h!8 p 1 n n\"$&#→%('0∞) . . n e = 1 + q = . 1j j f(x)m f(x)m g(x) = eg(x) ù ,›2x – l–†|‰| –}˜ ‡!‡¨0 |‡l| x ©l– ˜x )l €7€ˆ}–‡ ˜ s ‡ —%ˆ V‡ —€ }„ ‡ y 8 | y ‡ s I— ©0€  l h› — l )Ú ) € s ¤ ‰w l f(x) x l „ —%‡ x – y †– …¦ l | l „ ˜ ‡V£

1.6 CONTINUIDAD UNIFORME 35 k ˜ q ƒ„  ‰˜ l ۉ|g–}D – ‡z | ˜ l d„ | wz › l €7‡ 's l)x€w ‘| — l¤ ‰w l ‰|© ‡ l Ÿ l €‚D}– D –†‡ ˜ l „› ‡ Yx y €0©€c¤ ‰w l e „ {}z › „}|1 e= „ {}z › (1 + x) (1 + x)1/x = e x→0 x . €7‡ l | y ‡ g| x→l©0“x ˜ l ¦ cl x l 2€ „ ¤{z › 1 „†|f¤ú Y½(xÇ ©À)$½ ∼ÇX¼gÌVÁ(xü Sº)DÌ ¿½ X¼ „À©Ì ¼ ÁaÀ™¼X⇔DÀ Ï ™ l „}| ˆx→0 x (1 + x) = s … l 7€ †– eÛ  8z | ˜ ‡ x lhs l | w| l | y d‡ €‚‰| ‡ ˜l s ƒ„  equivalencia (1 + x) ∼ x 1 0 !h8 f(x) = 1. Lp‰| ‡Œea›¥rsb(lD¥$zxfo´yu)l€7ln€‚¤–xi=©–ªccÛe• awi©Dxeog|8sx”x n|0l—%−}–e‡V‡2˜ws€|elf| l©Û‰h−l xŸ|€xilpllsh›,—|eel—gnrs € b„}Toy‡‰o´‡Vxê h€7ƒlPi}– i‡¥cpa(5 ˜exssrl)b€y=•©l o´d‡ zxs€‚l—eui› clxsow +y „ƒis2–†n…cex v8o−›e› sxrelw,sn| §a”yosl I— h)%— €iV‡ pl €¥ 0 e(– ˜rxb) xo´=l!icƒŒ ƒ„o¥¥ § tangente hi- g(x)x→a x x• = ex − e−x . x ‡Vz €7› ew xƒ„ +xcey −€7–xu V‡ £ ahghf€•l˜ˆ©lˆwl−€|€x™r−‘liilDlg|ppg11l‡d00l©0•••••l(uee(„H––€7–xYx—„llxrr‡7€myz–})||x—%ƒŒƒƒŒ)–bb`‘˜|„|t–}˜y7y7|2¥¥=V‡e7y¥¥¥=o´o´lult‡••7y••€l((–}n+((2wwllwwlsv|h−a€”ii}„−alltt€”g||g|g|x¦…ccg|…|§y¦…|xosx0)0ooDl€0s−xyA8)2p––}„–u)€–l©Vsˆ‡‡‡‡d}„x5‡zbzzxz==b‡Yx===Œ©lƒ||¤||e7y)+ˆ(xY)¥gwls¥y‰−−[ƒh=1fŒg2ƒ=l©0llzx¤·(aŒ}–(¥„x1(¥,x¥|xŒxxw‰x˜wŒx+,xƒ=lt7yt)¥tw)x)lE=→1l¥l2„∞a¥‰Ûe|{}—I€)zx=”€=}„›a−=Tn+ll©ˆ¦…|g|1¢)aI—)0xxT∞8€ˆgƒ– ‡VsŒ)˜1ƒjp8x}„ƒe†–P„ƒ€ˆ‡¥€”¥xwqŒ›¥›„`0¥n‚€¥P¥”€‡bx–+5l©§qxt=0lhb…¦I—˜•x|Ótxxwey5s¢‡)il©©l€7©l·sjg|phtl©€xxPHs}–xƒ‹h=¤)Ú0xjƒs´ex−€0§Œ‹u0i–¥€x‰wD2ru5‡‡¥1p0D‡z¥s‡(4l€xb−w|‡Q|0(‡−Œ|0eRs|aax‡y|I|wo|¤)¥y∞xry}–)˜|–}yŒ–}yl1e|l–}‰w|bŒ|h|x|†–a–}l¥=q=|wl¥|…o´wf„“w's„†§uwƒ„Vsblwxlbf‘§8yxx2›€i˜u221n1sq§›!→csxx„l§sxj==n–}{c−az}„→—|‰§#ll™x›l©„−§|isc{¤„›zˆ|Yxl©‰w‡o´›+∞yx0˜‰iy}–xYjc©l›„l)xlno´‡h›‡s7€2∞yl511ŒxYa|x→ƒ–¤x7€n„|‰y−+–}¥{}=I—tz–}s“|ep„›t7€t¥Œ›ƒ+‡¥yxx(e¤–)i8a¥(yQ§„›2∞‡¥nm1xm€2˜8›yra—%x˜sa}–a›x)©l='…ˆbƒ)e|p=l©f‡ŠD›rqx)–a|ˆ€l===V‡a¥liW t=|¤ary7ll©||r−axl|gYxy”w‰yˆ€20ltˆb+yrl7yŠ–}ƒ1uulˆ7€|gŒl=PT∞€–¤m¥ˆw¥uxDƒ„l§x€t2{ˆ[yzD¥mla0sReŠ+€8–pax0c,lx•ln›A§§t©lƒ→∞–wea2|D„ilˆ+t!x{¤rzng|–πlry7¥|›o+l)l©lc“l|ˆŒa0ty)xYxY7y|Œa∞u–y7ydlh‡oyl!mˆl”y¥‡l€‚7€zˆn˜|fle|wŒ2–¤–}˜lde‡fl0¥|l[aeyytle1€–}c‰|88r|xr|h5lˆl,uo5›‡›ie0…+€zˆss„›–n5=ˆl)lllzaee∞€gÛc|||–}nndxŒ|e)iy””y7yl©1)ooaawlllxˆss

36 FUNCIONES CONTINUAS(fTcCyy„}ƒ„l©¨Œ ‡x))l)xoleil©x–7po€„„nx m˜l©¤o`€xo–n)l‡V¤x†–wgzrwnw€Ÿtvtep|´8‰wJelll¤›hih−|g>knDal¥ln›hDmw‰҆–1˜„›cDg—‡d0±•†uƒl=¥—glesV‡–taVr0€J„‡a‡ƒlzV‡7€}„okl7­ ‡4|x}–›x™2”€²'¢|s{|ƒ}„ex©´nˆ€cS.˜·‡¢š7y‡Vl%—q=n}—I1l–¤›—−Œç‡d˜›„”€)y w‰¥P–u€u¦…l€ˆ€§lby©lf–‡Œy©82|n−x¦g5nxY–}¦%[k„†H–2y7©l„›§ay‡a–ls%x)ƒxcwB|‰vx,‡¥lYŸ<ol©¤„Dˆ‡0b§‡D¤)8xRV‡m‡s‹x]D€w‰ls||n|lHs–1˜–©”€S„%lH‡yp|t0yz€ˆl§‡l¤}–|†–›!‡ε§w˜a|0|A|‰w„ƒm|–†˜e|cwxw€l©x”7yƒ‘l‡„ft€§ws˜l‰wJ˜(oxu>V‡‰|c•klxuyllw€v©le˜––|ln2›)|⇒llxg|0r0g|¨‡–}€||)†–w|‡D‡r…¦[© l˜−a7y|™y–˜€a|Yx8U‡ls83lzy‡l,y7d”Ÿg|–}€€ˆ2“„flb„f˜|n%—l©odwlz(|‰˜ly7w¤]xv–†V‡i0yŸ”g|ˆ|–lf‡„Hw‰|‰–€!jlno˜”€‡‡y”Sxz˜l|–}}–¦…ll|r)†•|¤„ ‡˜xd‡)7€[f=„emƒ7yI—a…x}„‚€n‡E−>˜l€c‡)›,8D€”l©ekSl}„ˆ€b„}‡ε–†¦…‡x”m‡Š4|l m|w‡]l8yx ›g۔€§y)„}„xYe[©=‡—‰yˆ€€7ˆ‰|ay„¿ln‡€d7€–|,¤˜[w˜y˜t‡xV0Jab”yw‰‡e‡‡g¦„D‡0l›],qxl0y„ b§n¤lc0–xéy'‡nz›!‡]zgwlo˜ 8xl|s¿z|k‡Pl‡‰n7€D—%J‰|syx›„‡ˆtx1…'>`l–†‡‡ryi¤| –}–n|y¤l‰|w‰wlés©n˜l2uw‰||g‡l¦g‡Sklulauxw„l©¢ž−„ƒ‰wl©x[‰™|x¥ž¢a˜5x1nl‚ž0l,|Dˆ,0¥x˜§‡bé‡ywyl„ƒ|ˆ€|]d¦I—„n)vs8y‡VyJˆ¥}–D)lt}–x€k||‡Vw€ˆ|Pw w|2¦|wCDje|yéšB)‡VŸ©§8z[l©†–ƒwa}„¥0ˆ|e€Dx– d|––x‹,x¢ž‹‡‡£l–zwzyb©lž¤pxŠ||||‡xž£]ˆs(„¤ w‰xkgw Dln−>l u k1ÿ‡¤„)Hƒ  kk‡−>„x→¤{›„zw1› ∞¦‡ DxV‡fw B|ƒ(x …x©lnlnx k€–k‡uVz−)|l =y(ynfcnk(ˆ ckƒ ))<k§ −>nk11„k→{¤z y› sV∞8 x ›qfl fg¦˜(–yl)lz˜n|fwk0)‡l =B|s )… f—gl (€}„c¤–u ©)l 8Bs |‘§ ˜ clR‡ |ˆ „ƒy  ˜ l©x – uw 8„ ˜  —˜ y 7€ –ƒ) | u w ƒ„ ) € s ™‡ g|l 7€ ‡l©x ©l x 0‡ | y †– | w  l s f | c}„ ‡!¤ „ }{z › p f (xnk ) − f (–}y©n0 –k‡z )|tq 0 ‡=|¢0¤ ,w‰l ƒ ƒ −> εm I— )€ˆ y‡ ˜ ‡ kˆ ‰w kl—→l)∞xYy‰z l t| 0 ‡| y €ˆ ˜ f (xnk ) − f (ynk )E—%yy•„}‹ ‡wˆ€}– ‡|„lx|g„}w€y–}T0¦…©l|B– )xe‡s‡ˆzzw}„|¢o—›„V‡t„ rgw€y –¤0„©lel©©‡x—˜l©mDx | YxVs‡x‰˜y”ya˜ –}weul›˜|Hl ˜lvdwwlw–¤ŸYe)ls|)e|V‡ xY¿€ €l”ya0 ©l„ƒvl|–paDxl lpsr€w˜—yor|4l‡¨{zxwo7€u }–†–x|gi€ˆ—%|…m8¤i‡)”y m•‰wlg¦}„la}–€”ls–}‰|ca…}„ ¤–‡i„†c8˜o´‡'‰w›„}„i‡!©no´Rl ˜–}8n‡˜„d©l…lp˜xDe”€ly €ˆsl€o lW|—Il˜2›‰˜iy)n7€‡leˆ€‡‡o´qi§x0¢em˜à„¤r4dàlh‡Vsi©zx0c›2tw‡ sa•ryw|‡a g|gxs˜xllDl0sl€”‡ou}– €7‡0wbV‡x|‡€0la€™|l©5lvx—)x ¤I—–¦w‰˜‰uw‰)x©lw‰ll€ˆ‡˜7€l„†–}l„|2w…‡w|¢y)x©|e`‡¦g©g—…¦€„—g‰—87€l©•‡7€}„x€w‡V‡¨lˆ |g€¨¦Û–}©l!›0–}Ÿ”“x›!– ‡dz˜˜ )u|0 vl‰‡€v}– ‡0ˆw—%‡| e|Œ|V‡©l l€x£—I€—uŒHƒ ll)ˆ‡‰z•|€ulٓˆ7yls€ˆx©ll€ˆlŒDx7€8|gE¢©|8s†„0– |lYx˜ƒ––E”y|˜lêx l˜}–§w©lˆ€Š|E8x `ll”xg| j‰—x|0‡„ƒˆ€7–¨y‡Vx ¸07€DI¦‹5—‰‡2¹‡ )v„€Š4sg¦)wgs 0©¢–}l—‡„ExãU3I¦{z‰w|h‡Üx ™'y©lŠx‰ ¤–¾–'s˜©s˜jó™§lA——l ) w‰€0„„z#¤Dl©l©lu l‰w€sRxcˆ€¿l x¿[I¦©¢|e¬i0l„7pd‡¢©,o`˜‰€‚ãxYj1§–y—l]u‚p)4§ CIq}–„ƒ€ll|e׏{ƒÄҔxt|z|wf©rDx ‰— ±‰s©„©l€7„ƒ‰w8`§‡Vx ™›uˆe— y€7©xe¦–¤)«–†–€¦guq›†–i¬‡!–†w)r˜˜€g¦ n”–l‘l0l–n ›2ƒ„lbxi²|z)wD­|−−7y€7‡r©w|lE–aaYx¬l˜ y|}–˜ˆ€¶R„|l ×ed˜l|‰7yýq%02›nˆl ²}–rD€”„¹n©‡‡z¬‰w…s|¼x8u y's}„))2ˆ€‡y€dó™‡ 0['!›˜‰a–ê‡Çlz,%(€| fblD˜xþsvl]%s !›˜lˆ´€‰| ˜|çl³8E‡7­l—¦eŠù|²Xy†–çҀl©s˜‡ l©x8˜AyYx| y ˜ x x

1.6 CONTINUIDAD UNIFORME 37 PT¨ (ep‚x`oo)r8´ yeDn)mÒ„ 0± ¤adr ‚­‰w B²3l´)·.—qƒ “fç Œ ( –x)f© :−€ˆu[0P©,(1 x˜])→ƒ <n5 l©x xs%D‡VlP „d| [y%— a}– ‡| ,„†wb–}|‰]ˆ‡s ›„l | }– y‡ ‡V|e l©x T ¥  `§ w |4—%‡„}†– |g‡V2› –}‡ εT ε>0¦x ½ %Ç ¼ ã X¼ » û ÅS¤½ e ûdË Ç Ë Pr(687f(x) = A (2πx), Bnf(x) = n õk ž õ n xk(1 − x)n−k f köB1f(x) = 0, nöB2f(x) = 0, k=09B3f(x) = €ˆxCDl™£ d˜y ¥D‡l ¥X|‰˜ (©8‡‡ ›„›„C)x‡–}e|x&'P h ` ˆn[0… –Œ ,l´el 1€hsh]ims¤  w‰o0¥ l¦‡‡Vps¿|‡€ˆo˜ l}„inQ‡x˜omP‡¤ wgi[εol‘0,d>l1e„]0B7y)le”€ S‡Vrg¦ n€n– sly tˆ€›¨e)P‡i7€En–}r‡„˜¤ l‰w— y ©ll8ƒ„ ˜q7€“„ u‡) •¤ €wÛ¦‰wtzh|gŸYl ‡D—‰ƒ– €7‡lBz ‡V4|| n¦If„†f ‡!(ˆ˜ x‡¤)w‰−˜l4l fx (–wux|ew‰)¢ƒl ê<›!8ε| —Il €ˆ) £ 3 2 3 x(1 − 3x + 2x2). n j nkm xk(1 − x)n−k ƒƒ rrrrr n õ õ k õ n xk(1 − x)n−k rrrrr kök=0 Bnf(x) − f(x) = f nö − f(x)= (x + 1 − x)n = 1. y7yl xl€ˆ›!7€uŠ }– hV‡yq €)   }„„“−<ls‡ |¤fRx kgw n=˜©lk˜ l Hx=l0‡ ›xf0w rrrr fj‰|I—zB}–õx)†•δꀇVnk(7y€7xl©›ö )Dx sl ö − f(x) rrrr õn xk(1 − x)n−k. kö ¤˜Rtj –†w‰–†|¥'… l‰˜V‡– ˜lˆ€nk–†— €Rsl P „ƒ|%—  ˜d‡Bx”€–δlw (l|›!x„”y )¨lh §l˜8 d‡ |l‡€ › l | 7y l y –†| 2εw h j f™(l x}„| ‡)mx[0s {}z,|§ s „ l)w‰xx ©l– uu w‡ —I– lS) | €ˆδy”™¨l>„}‡©l 0xx ¯w 0‡| B– ˜ 1˜ ]}„¤– ‡ k [|e0 , 10l ]¨Bm 7y l I | ˜ ƒƒ @ H rrrr f õ k − f(x) rrrr õ n nö kö Bnf(x) − f(x) −< k”y ƒ ™lnkCA BD−x1 l—øüx¥÷ ƒy<GE¢– Fløδ| + xk(1 − x)n−k; l &A BD1 ¥GE Fkƒ I— ) €ˆq„¤q —‰€7}– › l €ˆqI— ©€ k øC÷ ƒ ¢ø n −x −>δ ε õn ε n õ n xk(1 − x)n−k = ε , 2 2 kö 2 }–u0 | ‡ˆ€ k˜8|ƒ nk|lCA DB—−˜2ε1 üøllx¥÷|–†êƒ |<E3˜ F‰˜øδ– ll < nk|ACy”BD−7yl“1løüx˜¥›÷ ƒ l<3El Føδn| 7y §™!l k˜ ˜ ö xk(1 − x)n−k −< ›˜ y7l l | kƒ l 8s §™x k=0 xxˆ ‹ |— l l| l ‡zYx }„y –‡ ‡ ¤ x‰w l ©l ˜0‡R| d x0Dl ‡u yw )–†€™€ !ƒ„z  —%x”©lDˆ€uw |˜ x w xÛew 0›!– l  | y7y )l q›› g¦l |– l zy”| l – n  V‡ € l„ ”y l ‡V€ l !›  ˜ –l Ôsl – l € Yx y €ˆ x”x S M y )„ ¤ w‰lƒ ƒ −< M T xP s  Dx {z ¤ ‰w l f(x) [0, 1] nkCA DB¥−1 V‡üøx¥÷ €ˆƒ −>qE3Føδ|‰‡ rrrr f õ k − f(x) rrrr õn xk (1 − x)n−k −< 2M ž õn xk(1 − x)n−k, nö kö n kö kƒ x ¤  l |gD‡| y | kƒ ˜ ‡nkCA BD−1 üøxI÷ ƒ −>3E¢Føδ x l  ‰w ©l ˜ €0© € xP [0, 1] § w y 8„ ¤ gw l I— ) €ˆ y ‡ õ n xk(1 − x)n−k < ε . ACBD1 ¥EGFkƒ kö 4M ƒ k üø ÷ ¢ø n −x −>δ

38 FUNCIONES CONTINUAS ƒ—l w¥ 8d|  ˜l R€‡ ƒ„ rr –−u w x– lrr |−>7y ™lδsI©l xYx l™y –†›¨y – ld |0l– s ‡!¦g– l | sg§¢x l1 — ‰w )l ˜ xk V‡ z (|kê − nx)2 −< (k−nx)2 −> n2δ2 n2 δ2 n ACBP1 ¥E3F CA DB 1 ¥3E Fõ n (k − nx)2 õ n kö n2δ2 kö xk(1 − x)n−k −< xk(1 − x)n−k kƒ k üø ÷ ƒ ¢ø kƒ k üø ÷ ƒ ø n n −x −>δ −x −>δ 1 n (k − nx)2 õ n xk(1 − x)n−k, −< n2δ2 kö § ¤ w‰l)˜  … l c€ ¤ ‰w l¦s —I) €ˆ n u ˆ€ 8 | ˜ ls k=0 S= n (k − nx)2 õ n xk(1 − x)n−k < n2δ2ε T xP [0, 1]. kö 4M ‹ x¿w „| ukw¦=–‰wl0l| l “ˆ Ÿ lHƒ €ˆ‡0 ¤–| 0 –}l—‡ „}ƒ„%— ‡s | l € xl R‰— 7€ V‡ ¦I) €“„ƒc – u w )„ ˜ ˜ S = nx(1−x) x }– |2›„–†ˆ€ ) €l „ „ l ¨›  x – | ”y l  é0Ìú†iÇbºº¤ú¼iºË „ºüÃÑ˺ “dY½Ë DÌÐÇÁ ø¼'ªºË ú}ÅSD̄»Ì Y½ ˼YÃBÊÌh}ú À¿º ü Ì©Á”¨ËÊ “»¼vg X¼Å0“ÇË ¼Y½Ãv”½Ã »2»“û ”½X¼¼Ê ÏÏÏ S = nx(1 − x) −< n < n2δ2ε x– M n > δ2 , „}!‡ ¤ ‰w l–7y l 7€ 2› †– I| q ¤„  ˜ l ›„‡ 4Yx y €ˆd 0– ‡z |4MˆŒ –Lema. n P’r § x P 5 sdx —l y – l | l  xy −< 1 (x + y). 2 S = n (k − nx)2 õ n xk(1 − x)n−k = nx(1 − x). kö ¨ 7p `o ´8nDÒ ±0Vr ­7k'² =©´ · q0ˆ  7€ –}› l 7€ ‡ x l™˜ l©x )€”€7‡ „}ƒ„ 8 4| }„ ‡ x  w  ˜ ˆ€  ˜ ‡ x ê n k2 õ n n õn k S= xk(1 − x)n−k − 2nx xk(1 − x)n−k kö kö k=0 k=0 + n2x2 n õ n xk(1 − x)n−k. kö tR¥ V‡ ˆ€  s k=0 C = n õ n xk(1 − x)n−k = j x + (1 − x)m n = 1. kö k=0 t —g}„ ¤– © 8| ˜ ˜ l | y – ˜  ˜ j js= „‡ „ƒq – nk m nk−−11m k n n õ n n õ n−1 k − 1ö B= k kö xk(1 − x)n−k = nx xk−1(1 − x)n−k k=1 k=1 n−1 õ n−1 xl(1 − x)n−1−l = nx. = nx lö l=0

1 EJERCICIOS 39á wgx 8 | ˜ ‡ ©l Yx y”l ¦… 8 „†‡V€ s n k2 õ n n õn k(k − 1) A= xk(1 − x)n−k = nx + xk(1 − x)n−k kö kö k=1 k=2 n õ n−2 = nx + x2 xk−2(1 − x)n−k = nx + n(n − 1)x2. n(n − 1) k − 2ö k=2d w‰lDu ‡ s ˆS = A − 2nxB + n2x2C = nx(1 − x)Ejercicios1 —%Œ lV‡ )€ | ˜ ‡x | wz › l €7‡ x € l ) „ ©l Hx ˜ – Yx y }– | y ‡ Hx ˜ l 0ˆ Œ 5l ‰˜ l ۉ| lRw e|  xw  ©l x – ‡z | (Qn) α, β I— ) ˆ€ Q0 = α ; Q1 = β ; n > 1. Qn = (1 + Qn−1)/Qn−2, cet5#Uǘ‰)þ }„˜ „ƒˆb)Š– x€ ‡Q| £ nÔs—I)l x) „ €ˆDl §¦s y ¢©‡ 㘠㇠ ns'‰˜ Pl º†r Rdˆ¹Q ¹ j  ‡ ý!› ¥0' US 3 3˜ ‡ ˜ l $„ „}ª– ¦g€7A‡ « )´ %q ­70± p8ÒÄp ¦r Ò p7o`¦r Ò †² ˆ­ nDs 1IhS`» Y Ñ æ 5ï RT R\"U¹VX¦¹ W #Uñ ¹ ™¼ 3¨14a3™IS ¾b2  7€DlzV‡ y %¦¤– ©©  5€ ‰˜ƒ„ l x nde| sw zi›gul €7a‡ ldx a€ ld) e„ ©lsx d%— e‡ x l–ay s–†¦… ‡mx êedŒ i– axs1 )€7› ‡Vz g| }– ) ‰s ul ‡ › lDz y s‰€7¤–x”©–l  y § l |) l€7– y £ › – , x2, ... , xn P 5 + 1 + n + 1 −< · ž¢¢ž ž xn −< x1 + ¢ž ¤ž ž + xn , n w ›„—gx„}1– lz |˜ ž¢ž¢ž © x˜ nq– uVw n x1 ˜y ‡˜ ‡x  ⇔ „†‡ x xi x V‡ |t– u w 8„ l)x ˆ  €7V‡ %¦ © € ‡ x l ) „ ˜  „}‡ ˆ a, b P 53 ƒ − ƒ −> rrr ƒƒ − ƒƒ rrr %— © €ˆ y‡ ˜ ‡ x x a b a b4 Œ l  M P 5 + ˆ  7€ ‡V¦I) 5€ ¤ ‰w l ƒ„  x w  ©l x – ‡z | (an) ˜ l ۉ|‰– ˜ ™%— V‡ € a1 P 5 + )”€ g¦ – y 0€ ©€7}– ‡ Q an = 1õ an−1 + M ö x – n > 1, y – l | l „ {†z ›2– y”l‰s § ))„¤ w ¤„ )7€ „}‡‰ˆ 2 an−15 ž l)x ‡ „ª… l €5ƒ„  xvl  w 8 D†– ‡ | )l x ê ƒƒ ƒƒ ƒƒ (a) 2x − 3 = 5 (b) 2x − 3 = x + 1 (c) 2x + 3 = x + 1 ƒ ƒƒ ƒ ƒ ƒƒ ƒ (d) D3‡ x − x −xl x+2 0 =‡ x 5 (e) xx l −| 2+ x ”x−l | 1= x=−xl3| (f) x − | x= 3x (g) (3x) + (2x) x.

40 FUNCIONES CONTINUAS6 5e 8 „}„ƒ©c€ „†‡ Rx x – uw – l | 7y l©x „ {}z ›„– 7y ©l x ê (a) „ {¤z › · mx−1 (c) „ ¤{z › „ {¤z ›cp x + (x + a)(x + b) (b) xp/q − 1 x→−∞ q x→1 x − 1 x→1 xr/s − 1 „ ¤{z › „ ¤{z ›c(d) p k (x + a1)(x + a2) ž¢¢ž ž (x + ak) − x (e) c (x − a)2 x→−∞ x−a x→+∞ q (ax) „ ¤{z › „ ¤{z › x ”yl u |(f) x3/2 p · · · x (g) x+1+ x−1−2 (((nkh7€xYl 7yV‡))|)gwI¦ y”x˜x)xlx„„„→¤{→ƒ–z{¤€5→{¤z→zs›)›› +S0R€¤01∞‰wxxx„¤1xxh→™l„ 37¤{z−+−› 0bx −−x–‡−x”x0 2|lfl‡32f||y xxx(©l}–((x|x25xab)w ++›2xxˆ(– )i)˜ V‡)21 |(˜4lx(‡„z)n{¤→z©y˜› ‡)x0l |e„→{¤x0„ƒzu„›}{‡→z0›xˆ+x 1(€1• alwõ1−0xg|1–+y7)l00+u|}–−‡D2‡ ”y x‡(|lx1(0πxl©c‡ö(lxxxπx| xê)()qxb()(xam),(ob)())jx)x„§‘x→¤{→z ›→x„„©l00}{{¤z→z ››πxY1y‰/0„z2−y7((8u xb0Dlxx‡‡ )xy−xx( x31˜−x4l−|y”xu 0xw ‡x— x)l7  x)8 › ‹ €7}– d‡ €7£ xx – —s(a) f(x) = x ± – ˆ1 − x x P/ ± xP d e f· · (b) f(x) = x − x . (c) f(x) = 1 g1 ‹˜0`‡lx „›%¦ý ‡‡ (Ú)q¹dl©Ä)z rp)xY€Hy7¹pfl„¤(sVxx“§™)U3u y€U' =8)¿zó›™Ûg‰s» )¦gó¶rrrr 1 – xlfz+D­„| ‰˜ ´Bl…x1®V)‚±„ƒrrrr€7nˆx}–—pxf‡ ¿xx´ – hu˜ Ø wlÁ ¤p t× f° )r Dn iÄÒ ´)± x9 ,+– „†°5l ‡x|b®Vxl y7±0D©l© p2‡x—IxxÑ)• 2w€,¦yrxg| xÒ0.æl˜ –}|‡p7‡ `o|x$j ©lx3˜¦rx lmiÒê i²„ ‚­,Xjxn©t – sau „ ƒ€wu0w–‡8l |‰xq›| ‡x7y¦‰l +˜€7s–l l©˜ bx}„u ‡‰yx”l8“xlzs|| l Ÿy l ‡‚€ !›0ä'–¤ˆS0Š }–˜ ‡ ‡ xDx s ¢©ã x (a)0 ‡ 7y u (x| 2xxx,),x”+lx 20x‡ xx,l (| 4xxlj |)x12,m x ,x l 7y u 2 x x, x q (b) x l a | 2 x , x + x l | , ”x l | (x2) , x l | p π xl , x2 xl | 2 j 1 m j 1 m 2 | . x x s t c(c) x 1 , ³ x´ , ³ x´ + x − ³ x´ , x2 − ³ x2´ , 1 − x + ³ x´ − ³ 1 − x´ . xx ¥j 8Š†„ „ƒ) 7€ €7ƒ„ ‡VhI¦ 0) ‡ €–| ¤ ‰w l ƒ„  0 –}›!l  8w „ dl©0Rx – ‡0lz |¨ d x 3y ‡ =x ˆ 3x + 8 y – l | luw e|  w z |‰¤– © x ‡„ w 0 – ‡z |E€ l 8 „ es §10 ˜‰l 2 j ¦%Š  ‡„›„– x ›2‡ s —I) ˆ€ q ƒ„  l  w d 0 – ‡ z | ˆx5 = x + 16

1 EJERCICIOS 41 j ) Š“ž ©l x ‡†„ … l 5€ ¤„  l  w d 0– ‡z | ˆ2x5 = 2x + 1 j˜ Š¿l ž w ©l dx 0V‡ – †„‡z… | l “€ x l¤„  l +w d00‡ – x‡ z l |  xl  x+l 0| ‡ xl l „d }– |xy”=l ”€ …¦28· }„ ‡ 2 ˆv8u ƒ Rw 8z | y  xfx ‡ „ w 0 }– ‡ | ©l x“y –l | l ƒ„  x x =c (0, 2π)w Xj l “Š ž l©x ‡ „†… l €5„¤ l  w d0– ‡z | ˆx2x = 1 j •7Š“ž ©l x ‡†„ … l 5€ ƒ„  l  w d 0– ‡z | 2 x xl | x=1l | l „}– | y7l €”…¦) „}‡ ˆ(−π, π) 3 Xj u Švce ) }„ „ƒ)€R„ƒu› l |gd‡ €v€0 }{z —Ú —%‡ x – y –†… ˜ l lw 8 D– ‡z | y”u Œ l  c > 0 Q ‰— 7€ V‡ ¦I)€R¤ w‰l ƒ„  l  w d0 – V‡z | ƒ„  ˆx = x11 x − nc = )€‚ y”u õ n x + nö yx w– l l)!lx }– V‡w | e| l© x w z |‰–¤©¨ § ˆ€  †{z 2Ú € l ˜ x „ †{z ›2– y7l©xh˜ l „ƒ x u‹ | 0 – l € y ‡!¤ (xw‰nl )x – f j ˆ) „ Gj Åxxnç « I— †² )rhˆ€ ®V‘±”±”©²  ˆbŠ  n −> 0 ˆ ce 8 }„ „¤) €u„}‡ x ©l x xn m 1] n → [0, 1]12 : [0, (a) 2› ‡ | ‡ z y ‡‡|e|eu ˜ˆ€l lˆD€ –ll | y7l l| sy”lu‡ s g¦ – l | (b) „› ‡ | ‡ z y 0– ŸlYÒ w7p| ”±„}y}–q%‡‡ r¦|g˜iҁ ²ill)´)€x%q d  r©l0¡ ¨¬¡ –¡ÑYx ˆSy7Šr¦™l Ò wæ |p7`o xr¦PiÒ i² ­‚[0n ,«1]´)qoy )³“„ )p¤ iÒwgX² Òl ²†)´fq(xÁ)´)„=± ƒ‰q&xV… ² w7p 7± nj ’²X ×‡Ò I€7‡VҔ† ®B— µ‰‰w ‚pl©qIx yÒ ‡©n sel  |‰å™æ p€yG ˜‚u€ m©l eq Dx ×s ‡ |13 [fx 0–(€€77u,x‡‡VVw2)I¦I¦]– =l))y| „€u€ ) y71„©l¤¤l©−x5w‰‰wx l¨`lxx¤ ‡ w‰ˆxx| –––l ˜ffƒ lx: )l1[‡40x r−,0ˆ21¹ ‡x]Ílj ò|2→y ƒ }– |=w[3U0“1þ ,s 1] l©x 0‡ | y –}| w  s l | y V‡ e|  ©l x S x P [0, 1] y 8„“¤ ‰w l14 f‹l (€ Yx2y'7y )zll “sbˆ Š g—l |7€ V‡y g¦‡|g„ l  S § x1„†‡ ,xx2˜ ‡Px l§ | [0, 2] f(0) § =ˆ l … j ©l x « !›  f©r (¬ix­81V® )¬®=Dn s‰fl©(˜ x“ˆ 2)ž Tφl—I0k5v‘SŒŒ |‡xxll)€7(yl0|yˆ€‡Vx}–y7‡P„}Py8¦Il)}–f|g}–a„)Ú7€)|l[=[›2)000€!„}w l)€c‡‡,P,†–xfx¤|11|I„ƒ(‰w)y)f©(xl4–}vxˆ04l€)||y)l ,l8}„wx−x1‡l„[–„—)0xu0w f¤‡,fld€‰wly1|(j|0lŸ8xxl]y}–©l5„‡}–[‰yx+f|+0|8¤(zww,0l©|x‰w 1n›h1yB‡xu)l]‡d)mwg——Il=€‘ˆa|„}„})=˜ww‡– u$€ˆlflh›5=8’|We|–}fj›|g—˜x€w(§ x}„x‡˜l0n1–z+zˆ)llyf‡7€ 7€|f(x–+Tx¤–nx(1y˜©l0n‡)¤m—‡f)‰wxRˆ=‰w(lf!lPy=l©Â|˜j(c)˜Iõ0ly rx)l–‰¤„fTlƒ„ ˆi‰=(I—|˜x1§Dxl,)l©ll)(©‹fy‰˜€ˆ}„}„sv§D(†–)ª–|ew—z1—$ €§‚P00x)wxl©Cl‡–†Bsl4|tY€y|bˆ5§`  8Šyaxx›0——„0€–x22– §©llHƒφ¤z €€¨+gw‡y(f}–‡xnwl|!› l))bye|xcfBx–P22¢=˜W(ˆPx0lr •=0‡)€0w5jˆ©|õIg|T€5=ˆyW1‰0x}– „ƒ|–7€‰iwq‡ z‡Vfw˜§DŠ|r(I¦•l©( xw ©§D€|g$€Rlƒ„+l|0©C8¤˜– Y„‡aw‰‡z0  |lf)xs15161718

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2.1 CONCEPTO DE DERIVADA 45d2g…¦l–}†• |V‡d€ˆ©wu7€˜y7€¨›!sƒ–l –)l—‡8d˜z¦g¤ ¨|l„HV‡lvw„ |©lq€—I†–|sx˜„ƒ¦… )ˆld‡–) €ˆlxhy(„4|l€ˆuy˜‡|`y7‡v„ƒlf–†y”|‰•)l0—Illx–ƒ¨˜=€©)!—‘lsld€I| |gy”ewui„ƒl’v0fx sed–ƒ„‡yl)r˜v~c„—%e˜l©l+‡yV‡nf„xcl ‡yv€5c•ƒw„| 7€dli8‰—z|ea–s5–ª„}u©l7€¦e—l–}Q˜xew|‰›§w‡Vs„}–}ԇ€DÚlds c˜ƒ–€ˆ)‡E˜´aulˆjv|el|eul‰vd…}–c 2g|‰wmllxu˜ˆ¨Ú†–ˆ=lg¦€ oll‰|2g›–}|ru–}‡4‰wdÚ |ld¢©li|‰|l©vjfäV|v−xye¢©‡ày72‡äryvl˜x8edˆl20Dä8nu}–‡–ˆ£5t|v’c‡|z¢BÛgˆÜ|ix©l©¢a{|‰zls¦äl„˜– H”y5sŠ%—yy`l©lV‡%—‡d–wxl|€d˜‡–†|e„››¨sd7yl©ll lx–†—%xŸ)xg¦|g)–l ‡„ €”‰|h›l©0€”x€7–ƒx©}–tl©l‡Úg—Hsu }„0}„–˜¤„„˜‡¥‡˜‡zfll|s`‘xls¿|l©d˜©l—˜Yxy(8d„y7l–†lluÛel|0€ ¨›©y'‡˜s –)880w˜zz)¦I–¤„„¤v€ˆ|l¨)˜|wwl ˜€ˆ7y©l„}}„=„¤‡‡‡lx|‰™ed‡$l ty7€ ‘  ©A‡ g¦ '& „¤„  fl ()› Y  lw | dv) 7y¤¡€ §ly 'ss¢ ¥—%„™V‡l €©—I0x–w  u –†}„l –˜ ˜ ƒ€ ›2˜d wBs ‡ l ¥B| ¦§ lsf„y ƒ ”y 8wz „¤„ƒ w }„ h‡ )W g— S„}ò8¤– ©§D(© D§˜ $„‡ ©C —Y B  „¤ƒ¡ §©x Q¥ D– l §!g| 0ˆò –ƒe $x DC|e £ y”$ w ©§€ˆ¥‰8 „ l© x ˆ V‡xz | ˜ hˆ – ˜ § ˜ ‡ l ˜2.1 CONCEPTO DE DERIVADADl „de„ {}fiz „› Œn–liy7cl ifo:n(eas, b) → 5 § x0 P ˆ(a, b)  R • w |g0 – ‡z | l)x derivable en x0 x – Dl ¨ – Yx y”l f „ ¤{z › „ {¤z ›f(x) − f(x0) = f(x0 + h) − f(x0) . xdy„l†{az80x©2›e„g→ˆr¨f–x)iuy7x‡v•©l0“nwsxax ‡|e„ƒc•d—wsBi0ao´|g–˜• x‡VswnzD–¤|e|–−l‡¥ fza0f|‡ˆ–xtV‡:ze0f„|[{}zra„›xalH,–lfb”y˜eV–l]–¤s: →x(ll la|hd,€ →5ealb‰—rx0)%—il©€ vx©l‡Vax€H−˜‰l→b’ →„¤|l l7€ye–†5˜…el)%— nf5h€¦gV‡ lV „(€(2lxa¥f),xV‹| (bxx)0l x)h–„}„ƒ„§„})‡ ¨›x l©l xd „}lƒ„l)|a¨› yd‡ e˜ lr‡aivxdaePdra(iavja,—‰bd7€ )–}a›s8¦§dl 8s eˆ€ lV|fŠel©Yxny7l – §„l©™xl | ˜ lb€7–†—%¦… ‡V© €Hg¦ „ƒ„l2—l–}Ú©| ¤ w(a– l ,€ b˜ ) sh§f©l xHl0¨ ˜ – lxYy7D lª– €| sdeƒ„„}‡ xxmI0 n‡˜ t|„›el)rx lp–f7y„r¤„+Vle xt™s(lapal2›c)e|i´o–nln=lŸd„“l gi—eheOnow„→¤{zXm|t› 0eyˆ´e+‡ t˜rfil–jc(xaa˜l0ˆ “–¤+,w f¥ (dh—xhƒ0 )0– €w‡−z)l |8m  | §f(a) f−V (b) = „ {}z › f(b + h) − f(b) . sy ˜ ˜w —l ‡'s |el©©l S“xYx y fj'˜™Vzwl(‰|€Dxl¤––ª0€© )sy Š¤„ = yry8em8|c| utsulal|ƒ„  h | hu →€ ©0z Ûe−© ˜ l l ‰ww € l|gy ¤–D•ƒ© ‡V– 8‡z€7„|›¨s§f 7yt7yl—a—l n)l ˜Dxgsl e„— n8‰wzt|l©euDx sw |‰„}‡‘„ƒh‡ ¢…¤ • y − f(x0) = f V (x0) ž (x − x0) .a wp§ r‡o‹ xe“„irmr—%o‡ar}„c}– |‰il©o´‡Dx ns›„I— }– l‡© i0€nq˜‰el ©al‰— ˜ 7€—%}– ©› h€ˆl¨ €l }„| ‡u ¢|¦… ˜ 8 ‡0„}‡dV‡ T€”€ 1l©€ l©x™(xx ˜%—)l‡=| „ƒ˜`f– l(• wx| |g0y”—l)0– +l‡ z || ©l xu˜ l w x 0€  yf‡VfV (7€ x‰|l 0|‡ )˜w ž l|(xl s−| y ‡Vx7€0|‰)‡ e|  w 0 x0 f(x0 + h) − T1(x0 + h) = f(x0 + h) − f(x0) − f V (x0) ž h = e(h) ,

46 DERIVADAS˜ – u 8 „› ‡ xDs §‘x—l y – l | l „ {}z › e(h) = „ ¤{z › f(x0 + h) − f(x0) − f V (x0) = 0 .L|‰Œ !l‡an˜˜ d–¤hl  a→—l20u0•s¿l w‚Š | ©lˆ|gyuxhD‡ –|gw ‡z |e‘| l© x„}–}o|¤ hl‰w →p8l „e0ƒ„q¨˜uf• ewxn˜0g| a:0 –5 ‡ z˜ h| l l)x %— ‡j| ˜ – l | 7y l„l | y eh(s¿h)‰§ s™x˜ l‘l ‰Û˜ ‰|l –‰| ˜ ‡  y  l | e(wh|Q) 0=V‡ ”€ ‡€ ‡V”€ £ ½½½ (h) notacio´n ½ ½ ½(h) + (h) = (h) −w→’ →| l | 5 x l }„ ƒ„ 8!›  la diferencial de fy ‡VV (€7xg| 0‡ ) ˜‰ž lh0ê (h) 6 (h) = (h2) de xl—y – l l ‰w ©l x©s hlf en x0 Q | — I— ) €ˆ | h #dy = f (x) dx ‡f(x0 + h) − f(x0) = dfx0(h) + (h) . #f (x) = dy dx y = f(x) ˆˆbbªˆªˆ bbˆˆ ˆbˆbhbbˆˆ„ˆªªˆ{¤→zbˆˆb›bˆbˆ ˆˆªª0ˆbbˆ bbˆˆ ªˆˆªy”ˆˆbbuˆˆbbªªˆˆ bˆbˆj bˆbˆαbbˆˆ ˆˆªªbbˆˆ(bˆbˆhbbˆˆ ˆªªˆ bˆbˆ)Pˆbˆbˆˆˆmˆbbˆ ˆˆªªbˆˆb=ˆbˆbªˆªˆ ˆˆbbbˆˆb”yªˆˆªαbˆˆbu bbˆˆ bˆˆb((ˆªˆªhαbˆbˆ )ˆbbˆαªˆˆª0bˆˆb0bˆbˆ)ªªˆˆ ˆˆbbbˆbˆ ªˆˆªˆbˆbˆˆbbbˆˆbˆªˆªbbˆˆ ˆbˆbˆªˆªSˆˆSˆªˆªQˆbbˆ bˆbˆ ªˆªˆ ˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆ f(x0 + h) 6 (h) (x0) ž h f(x0) = fV ‡e(h) = ? ?6dfx0 (h) x0 x0 + hŒ l‰˜ l ‰Û | l ê l)x x –diferenciable en x0 S l: 5 →5 lineal y 8„¤ ‰w l f l |f(x0 +w h| )l −| y fV‡ (7€ xg| 0‡ ) −˜ l l0(hˆ )‹ xv=x”l ‡ (h) l c€ ¤ ‰w lx l l bvƒ‘ Á˽ ÊÀÇSÅ $½ºü ˺Sˆ”ƽX¼ Å©ÀÅ ¼⇒i¼ eà ÌD¼i¼eà »!ü Sº Ë¼Ê ÃgÇÄú¤ÌD½ ¼™Vû ÁË ÀÇ Ë˗I) €ˆ h |g0}– }„ †„ „‡ … … 7€ ª– Ûe©  l©vx ˜ –}• l € l |e0–ƒ©¦g„ ll | l©xR˜ l 7€ –†¦… © e¦ „ l l | x0 ⇐⇒ x0 . f fNotas y ejemplos e nl ‰„ x„ 0{}z „›, e– 7y nl to„ {¤zn› cefs(xf0e+s sh˜continua en vlx0w s)§ €¤ ‰w l ƒ„  w z ‰| ¤– © |20'‡ 0 – l ”y l w§‡1—%.‡ —x †– ¦gS}– „}i– ˜ f ˜e˜!s˜ dl e¤ r‰w ivvl al0b¨ l– exYy h) − f(x0) | ˜l y – l | ‰˜ l  0 shl)x ‰w l „ {¤z › h→j 0 ™ l l l †{z —‰7€ ‡'0 ‡‰| ‡ |‰©l‡ x „› 0 }––e|l € y V‡ €  h • ¤ h f(x0)m =0 ˆ 0‡ | €7‡ w e„ €  l 7€ 2‡ ‰| ‡ ‰‡ ê w h→0 f(x0 + h) − y }– | s — – x −> 0 l)x1.1.– Puntos angulosos: g| 0 – ‡z | ƒƒ x – x< xx 0l©xR˜ l €7–†¦… ) ¦g„ ™l l x= § ˆ−x | s — w‰l©x f+V (0) = 1 f−V (0) = −1 0

2.1 CONCEPTO DE DERIVADA 47111——˜ ...‰wll 432€77€©l –†‡...xˆ…¦–––¤g| ˜Pw‰Œ)‡ ¦¯llu— Dlxnn—•¨utwƒ„ oow‰–– xY|gssy”©l”y x0l¤˜ld‡–ˆ€ wl‡e|z8h|E„–‘| „¤{lz→tl©d›a|4–}0xv0en|g‡ g}„lr0x|„‡ |eli„ ywgvn| —†– 0axt| we‡`jbw xh¦g1vl‡›„„}mee|–}frˆs‡ t(‡+Yxzixecy )la∞n€ˆ|l)=:€nˆ  ifnV‡u©¯'0x€0gn–}uux´‡cl n„ƒsi|„ol np|Qj• wx1eu¢©|gmsnãV0 tꡤ– o‡w‰zxx |E––ˆlhxx‹Dx V‡‡„==W ||x y –}| w  f(x) = · x ¤„  –x ˜ ‡ x – 3 0 0, ‰| ‡ l©x“˜ l €7ª– ¦… )g¦ „ cl l | 5l s 0, ucw o– nl |tiy”nlRul Ÿal s„› e—gn„}‡ txoRl d˜ ol ¦Œ l D V‡ | x – ˜‰l ˆ€ 4¤„ t • w g| 0– ‡z | K(x) = x 0 −< x −< 1/2 §rx ¨l Dl ¨By – l | ˜ l  5— l €7– ‡z ˜ ¤– ©8› l | y”ls˜ l Ûg‰| – l | ˜ ‡ 1 −x 1/2 −< x −<©€01‰ © ˜ r hs x l  = K(x) Tx ˆ  K(x + 1) NP2T—§w˜ƒHNl.‰w‰|‡—7€©l‡2|Q(†–xxx…¦©0$¥))|%—¦‡5kg¦l‰w=V‡l|‰8„l©l„›€•†l—›2xY‡Vnly‡„¤l€7N=‡AŸ€7!›xt|l4‡x0›h•¤)|‰¤w2I¦ R€‰w1g—–}w‰ng||l¨„}Dlu0K‡“u‡–)l©|‰‡(|‘zzŸwx2|`|‡l n…'0—lD–8“‡xx¨wp|)w –”y|oˆyxY8w p–}y”y „†|8l‡-}– Ú)cw„H™o„¤)a¤„r•€`nw ‰sPlf”|g—‰|yV [(D8€0la–„d‰w0‡,|z)…l1| Q§¦Il]—I[ê0™ÚT©l,(0€ur|1˜x ]2l)›!s„}a‡=l8fx „}P(/‰|„}Nx‡†–‡ ”€„±)→}{zsˆ€ ›x dQ„ƒl=∞0€l vx}–{¤zE|T‡` uNe| ©€I¦)0(q)1z„xÛgxl©Yx)‡„x)y )… ˜xx˜ xv|l––™ll€7”y ‰˜xx–†ll0(Ûel ¨==0©¤„ T–),qYx11€0qpy u)„}sª– !‡€BQT‡†–Isˆ”€2—8x€ |sll©lŒxw7€T€˜l ‡¢{ƒ3zg|P/w„ —sDD‰| ±–¤.‰wª–‡g¦l.,©l˜‚€.„˜l)l‡‡‰l,xsˆf(a + 1 ) − f(a) = 0 T nP r y 8„¤ w‰l 1 n n > 1−a. 1™ l €7‡ x l  a n l „ ˜ ”x l©l–x ©y ”€– l€‚|‡V}„l „}‡ ‰˜ l 0 –}›!8 „ ˜ l s § —I)ˆ€ u© ˜  n P‘r xl  0hn = −0 = 0 V a1a2a3 . . . . . Q a . . . 0an+1an+2 . Vrrrr f(a + hn) − f(a) rrrr > 10n ž f(0 V a1a2 . . . an) > 1 T nP r. hngT2á}–|‰0 h›‡n.–V ˜1(|y—g0.y8P„}–)–}¤–›h|©lr=wu|‹%— ˆ'‡€„|‰0l 0l–„€ˆ‡|wx –}¤–|k5©l0¤ ˜x5y7wl l§¨ˆ€ ˜–€”l›¨…¦ll˜€ˆ€7)|4h8‡†–„}…|‡ }„0)˜l‡‡ g¦[€ˆlx|−‘„ yl!–†ƒ„1€”}–4¤|ˆ€,j w‰d1—%w• 0lw]–V‡ }–s˜|gc€‡ƒ„gD|e„ƒ‘˜–(8‡xz˜ „ó|ll))˜l |Rx€7l =–†Yހ¦…˜w%—l ©l|d‡¦g¥—g(l}–−0£„}q| –01Š ˜y12V‡ V‡l,7€ €7|1˜‰| ‰|)ßxx0‡Q˜––s s — w‰©l x f j 1 m = 1 § fp ² 2 =0 n n lŸ n q l ,w l 0‡ | x – ˜ l €ˆ ˜ ` —%‡V€ l „› g— „}‡ ˜ l ۉ£ ==„d‡z|e†„— 0‡`qpw l©•| ‡†–wvxy€”g|€‡‰‰˜ ©lx0ˆl˜– ± x P/w , l©qx y ) q› g¦ – lz | l §‘2lx˜ lxw D –ª¦g„ |Q| —0ws |x y– l4‡ | ‰|˜ ‡‡ 7€‡z †–|… )l¦g| „

48 DERIVADAS3”g¦l¤˜˜ |gw.–ll(l—07€‰|l |I)ª–‡…¦(ˆxŒy‰I)˜–Œ8)uV–z¦gll|gw)„˜é˜l2˜›ql™›!l ‡I–l¦gx8x›„|lI‰|–l(lz1‡–†—‰|fIÛ5)˜€s(l)P•‰w‡‘”xIYxxwl)lyl„”|2‡'I¦‡ ˜™é(©yq}–1l–l|)l‰|”|Q¤xy7(|–}‡(‰wlI‡l1y)€”l|eQ|)…¦w (8„ƒ)8Ifw„ƒ„}„)}„‡‰8¤P•šˆ˜xx™|ˆwDll ’˜”xu–Œg| l–‡w(l5w‰u‡€ˆ0|wylf)˜ 8˜(– ¤–ƒ„l˜)l„›I™u|)“xll ‡y7}–‰—˜—I|g¤l©„}x ‚€‡l)x0‰w–†–x7€0ˆ€„›lwg–†–}‡™ ¦…|gl |xllŸ)0–}a–€ˆl|‡g¦„zhfh›wg|˜f„ l–ux}–¤–P—g—}–|g©l©n‡l„})€x0‡|–|cc€„ xywg©li(I8¤o´™xx18›qwg–n§)|‡lz(lg¦y”||Iƒ„fl)–fy l7€€zl©e†–|•l©llx5‡Vsw„x €l©)lg|0dl)‰¥DxYxYƒ„0x‡Vyeyl–ˆ2‚€7€x‡|z–¤}–l©l¥|yaHƒx™–}‡yy |fV‡Bc˜w˜V|êll¥all)l„xs•l¤„we|0¤w‰g|l‡ w‰1Ÿl0I|l©l}–ˆye›„‡f˜ }– n||g—x ©lw˜l„†Ix‡‡ s ê3.1.–  u• w |gD – ‡ z | x2 xl | j 1 m x – x =W 0, ©l Rx ˜ l €7–†…) ¦g„ ll | 5 sdx – l | ˜ ‡ x x – x = 0, f(x) = 0f V (x) = x l | õ1 − 0‡ x‚õ 1 x – x =W s 2x h2 xxl |ö xö 0 −0fV (0) = „ {¤z › j 1 m „ {¤z › h x l | õ1 h = = 0. l0¨ – xYhy7l→l0„c„ {¤z › hs ˜ h→0 höƒH‡ „› ‡2|‰‡ ƒ„  ‰˜ l 7€ †– … f V ‰| ‡ l©x y –}| w  l| 0ˆ f V (x) D‡V|T44†•wl„}l˜ˆ'‡V‡€|E|xl)s|..7€©l—1˜ ›!80I8.P„w–„–}{ˆ„zlŒ„›Hu (˜‰|Œ–lazf–7ylhwl7y|l–fl•Vl−(|dxzxˆl•–0jwIu|)wδ}–˜ ùœ„}|eg|y,E|›l>Id‡‡0aˆ€‚–zls€—)‡Vx‰|0z|l5sw¤|‡0y”wg—|j–l ˜ffl˜‰yllw (|Vlx‡7€)l(x#| „ƒ→7yS!‡—gx8)alaδ)„}|0–}Š‰|}–=|ˆ)y7><P¨‡7yll)‹ l||–}I”€00„…h›˜0x¦…lŠˆlIx‡`y)€g—Ÿ+uls“T8}„ll„}‡‰ƒ„“„lx›„¤–h›y2x©ˆ|–¤!2l‡g—Py—‰‰w|x‡„}Œx¤€7‡llI|gl–V‡¤wg|f—gx⇒f‰wlf©l–(–Vjuhlxx©l(¤„x1l©wa„˜)fl¨xfm–)¿„l>•Vl©˜f(w˜‰|>€Yxxg|l7yy©lxfx˜)l7€€7x––0(0–¤–†lwa−>–…xx„}‡cs„z2‡y))0y|0==)zWg¦‡ }–T)l› „˜|„fYxwex¢lj00‘‡‡`yxfl xY,,xl|VyPll7€}–(€ €ˆy7|›h…xl©lBl(d)Hx —gaêl)I†{z0ˆ‰—−Yx<˜„},–€–¤y‡§€7laz©l€7‡'0f|‚€D–¤ª–+0ÅŠf–s…¦ly}„‡ '‡)%—T|8l©δg¦©›”y˜xV‡!x)8l¨„€lQ5ll„§l)P|XjD©lxYlz˜‰˜yV‡y”fYx| lI€7ll7y(| „–¤shlˆx5xˆ€)xlDy€ylIs–€”8V‡‡lu<D¦…x››2D!›–|‰–8l–l ‡fllD‡|¨| (||–”y ‡˜a˜zy”y”lxƒ„ |‡llll)Š xl | 0‡ x x –1 + 2x j 1 m − j 1 m x =W 0,xf V (x) = x x x = 0. –1(ž˜5t)ˆ0€€”ll˜l,€7€7›!–†0δl–ª¦I¦…–›)lê) |‚€0–˜szŸ”¤7y‡xq¤!lwg| ˜w‰jl …l‡l δ„› | >‡ (Dx 0s0,—δ¥ w‰)sl©§ Dx§ s ‰s‘—T x¤xwx–l¤w‰u | ‰wPwly 2l–‡nl(fx0|˜V (P’,y7xl!0δl )r)RŠy”x l8ˆ=„¤€7y„2 l©}„8 ‡x1„• sw ¤¤ >|g¥ w‰w‰0)ll –0¦‰‡ zfˆ€2| ƒ{Vzn1(¥¿πfxl˜V)8<l©l<„› –x δ‡0w˜ˆx sh› ¤D g—‡w‰„}|¢l„–†€ x}„l ¨‡l|Qs  0w w‡) ›28„ „¤¤f‡ w |‰¥‰– l2‡l q€„› — –}‡w‰| x™l)”y l˜ ˜€”lq¤–… 8x¥ l„}‡‡€ l €7–†l …¦| © l¦g›„„ lu‡ xl | ¤ w‰y ‡l ˜ l fw€–V|(ƒ„ 2xδ)‰| >−>‡ y 0 ‡ s D ‡|fVj1m =0 ˆ 2nπ 0 x =W 0x l | D‡ x xl |fV V (x) = 2 õ 1 2 ‚õ 1 1 õ1 − x ö − x2 , xö x xö

2.2 CA´ LCULO DE DERIVADAS 49# #y (x) = d2y ”d5–r(f5d”—%§•0xx˜x˜ƒŒ wf„ll™lƒ„iV..–(‡(l€‡eewVv(©›2|g1—2nV |:„}2)r˜x˜l0˜‚€8a)–†0.−)lqiIV˜cg|¨–d(¤––l|‡d–((vlƒŒ15l‡l‡V–‰|a→05z|˜˜|laaxYa)l¨D‘|2›w‡)‡7y)‡`(˜yfl)sbs¤xlyI8‡d!˜felsVe•}–n5l)x¤„„ƒ|w‰x‚€w‡—lrVe—)llf€‰|•–x©l˜|gw‰Û‰é‘in‰˜fxxl)wxw‰Vl)v‡s´eV‡}„d˜0©l|‡xlf(xg|l)ƒ„|•a„ƒsœPw––xx)l|ۉlwo˜xe‡0flDi„¤z)ld€7„¨›|)‰|–g|msxƒ„l|n(yu”xv†–˜‰|‡uanz…–P‘0T•d(l|˜‡˜v2˜sa)–}gw∞y•Ifx‡n(elwe1˜”l–ª–l(|g0lfVI¦…|a)πrsxcVe|˜r§(¨l()0P–0(B(l©∞iue)xuj|4l0––}nIƒ„vRxˆdn‡é‘Yxx2–xs)wc))I)a”y‡V=cl˜‘ln|wzx1eˆ­˜e(8lsj=|dle”}–l©πIe|(ry7l©sl“„z˜ƒ„t)n−x5fla(xi€‚iE¨¥m2g„›nvlv}–|ª–V Tx1¨›V‡…¦0x|‰V=)asa(ƒ„a•˜l•lV‡,0x)n¨(w4(0w‡e|dls|€1Ig¦˜x3)y|gg−e||.Px)az0)x)‡„ˆIPll)lDyxu0ˆm©l„ƒ4fssl‡VI|x––|gxQ§‘snf‡‘nV‡V|z7€x”rzDI—(‡e„¤D|j|nl|gdπ‡c|eyxw)sfg›P‡Vxj1‡™lx)las)ˆ€”x1|f•8u<lm…xxx‘wy|(y˜”!l<m|lnd•l©–∞|g–}0˜l(lw0§|xqde‡˜n0|‰©l|wfxx‡)wg|wˆ|–¤–a––xR)(axx‡˜lV‡|‡)0lz‹(––5y)lyxxalxljh|–„¤lI–}5x‡d¤xx|7€)|2z—l)| ==†–|f„}WwD¤nyx|ewˆ1˜w…¦f‡==u…W‡w‡Iw‰–πl©|ul008|g|fnsgP•wfŒ„l7€¦g…ma€y00|A,,l†–x”|sc–‡„lsB¦…a,,˜”l©|el©l™x¨lniÚs¥2©xxDxo´ln(–}©l>l˜e¦02|)lsÛg)˜f˜n©lYxd—‡P)xY—I8(”yg¦„ˆ‰|‡y‰Yxla€lq(ynll0et|y‰7€)–xl–Ifz)”€€lr)sd˜†–ˆ€ˆz)l‰—ˆ€…7y…(|ƒ„…clf§|ˆsl€a8zlŒ|˜ll©l|xy}„|˜a)„}l–2‡wx”‡lx‡‡lsH©l|2sx–l2–}l|f|˜x”|gxe|Iw§Vll•(¤„‡˜0(VDxswyl0–2w|€–l1{I„zli‘)l|‡y)g|lgwx”|)˜|n|‰0—%—a(x„l|gy(|Dx„ƒ%—5fi‡lV‡5wV‡}–I–†ˆ0}–˜‡|‡y‡V7€€n)||lwx‡)‹2‰|||§–¤€l—”yy|‚|iwl—‡©ll—l)x‡tylyŸ—%”–u|ƒ„y”Rx”€xo‡€xl}–ll¨†–l(…˜V‡|7€|xu¤€7h›(—∞)l)˜€8¢‡lwn‘‡|”yew‰•xY€‚l}„l˜—ewl)y)nw‡fla¤z‰|(|‰”€7€l(g|†„(›Vx5‰w‡‘…¦–¤Il‡V‡f00Il(§I)d|8l+,–lya|s}„}„‡‰|lle©eδ‡z7€©l‰‡ƒ„„¤)I)|||§x‡‡s)-xxxˆˆsˆ dx2 2–}}„˜˜ |h}– l|‰.… €7‡2l–†˜ „›…¦€ ql$¥x }–Bl˜‡w €ˆDx|eCl ‰sk52›„˜0l A©ll¨´0‡ x ‡—%x—¤„ Lh›V‡w“l | s|—%Cxz ©llw‡ xg|—‰›¨x€U0€7¤–l 0–ƒ–}I—8›–Ls‡„z l|—‰x§O)™€7€ €˜‡}„„l‡lw‰˜ ›„Du uw •©‡)w€E™€yxe|{z ‡y 0„ƒ„¤„› –}§V‡DxRx–¤|© 0˜€l©E'‡lDlxcs u0€7yR–†–˜„ƒl7€¦… l–|x—u€‚I7y˜ V‡ª–ul„V…¦|gl)xv˜‡|Ag¦ ›l˜l„ ©lˆ€l D•l0)5xzwy„¤„ 7€|el©§A–¤x™x0©ƒ„–}•˜‡VSwx |l ‚Š|e˜ „¿©lˆ 0l x–—‰}–7€ ‡V–ª€7˜…¦|‡'ll©7€˜€x©l†– …x l‡)„˜g¦l ˜lq›„ ©ll2wDxl s˜||eylƒ„„ 8€7†–„•…¦©l˜w x8lg| D7€ 0j–†–—%‡–…¦z‡z|V‡© | ££ê y(n)(x) = dn y”e½ˆú ”Ë ü ºˆ½Ê vÀ ÌD¼ ö ¼XdºªÆ‚xDÀ nº æ fPR— gw re|o§gy pl‡fao/sagsigcPjdeizl)o´n5xYnyehr˜ 1a‡wl.| e8˜ s| ulŒ ˜ dl x‡8 e‡| |gd(f˜aelr)§ €‚i=ª–vW ¦…ga)0cg¦˜ Ši„‡o´l©x xnx‡Q| • wl ˜|g| ly0‡€7}– –†‡g| … |)l©©l¦gxx2„ l©¤„˜xv lx lgÛ |•‰|w –a|g˜ Ds' –†§4x„‡ | lxl©–l| x y w| l |s y ‡V7€ |‰j‡ ˜ 4l w Š| s f– l s| gê l λf λ P 5 j ƒ– Š (f s g) V (a) = f V (a) + g V (a) j –†–†Š (λf)V (a) = λ fV (a)

50 DERIVADAS j }– –}†– Š (fg) V (a) = f V (a)g(a) + f(a)gV (a) j †– g… Š õ f V f V (a)g(a) − f(a)g V (a) gö ¡ g(a)k 2 (a) =˜¨ ‡7p |`o ˜ ´8l Dn „†Ò ‡±0xVr ­7… '² ´©8·šq„}V‡ ç € l©x$¥ l 8›„‡ x —%‡V§ € l Ÿ l „› g— „†‡ l©j –}xY†– y‰†– Šˆ8z Eê| I—˜ )l gۀˆ|‰– y˜ ‡‡ ˜ Dx ‡‰s x hl—y l– |l | ll „ l | y V‡ 7€ ‰| ‡ ˜l 0 f(a + h) g(a + h) f(a + h)g(a + h) − f(a)g(a) h = f(a + h)g(a + h) − f(a + h)g(a) + f(a + h)g(a) − f(a)g(a) h = f(a + g(a + h) − g(a) + g(a) f(a + h) − f(a) , h) hl)x hs0l0¨‡|—‰y€ l©x w– ‡z |4l | w a§ ‡uˆ „ }{z ›„– ”y l  w 8| ˜ ‡ hy–l | ˜ l  f(a)g V — ‰w l)x l©x }– | 0 (a) + g(a)f V (a) fPlx˜ –|llry€7|o‡ª– …¦˜pg| )‡ o¦gl)s„x lqic„ƒliq o´| •nw a|g20P – (‡ zJR|ts‰e§0g‡ g›„la˜— dl‰w Ûge©l |‰Yx yl– a˜  cla|dew n¢| a}–)| .”y l Œ ”€ l… 88 |}„ ‡ ˜ l gÛgfj|‰f(–J(˜ x))§tm l |˜‰©l xRl w€7˜†–t| …l )7€–†|–†¦g¦… y”„ )qll ¦g”€ l…„ l™|8 „}l‡f|(Jaa)§ Qs φ(x) = (g Ð f é I f)(x) = dy dy du dx du dx = 6 j φ V (a) = g V f(a)m ž f V (a) .¨ 7p o`´8Dn Ò ±0rV­7'² ©´ ·šq “ç  ‡V€ ¥ †– — ‡z 7y l©x – xRx l—y – l | l f(ja + h) = f(a) +j hf V (a) + oj(h) , g f(a) + km = g f(a)m + kg V f(a)m + o(k) , §Bδ PyB8ρf„B(ε¤0(‰w)0s2l )%— sIx ‡–˜ –u ›28 P „› ‡ BxH‡ xρx ˆgw(0x )y ‡V–sy”€–wl |„ƒ–ª2€5y ‡©0|g7€‡ 7€| –ªl©¦%y x }–| w –˜  ˜E˜ l ƒ„ „ • w g| 0– ‡z | f l |I—V‡lƒ “„¦)w €ˆy— 8– wl| ||h˜ ly‡ ‡P sa (S 0ρ) > k ˜ hl Bhs 姨(0x )l ˆ 0 f(a + h) − f(a) P h P Bδ(0) ¯ k = f(a + h) − f(a) j jj jj g f(a + h)m = g f(a)m + hf V (a) + o(h)m g V f(a)m + o hf V (a) + o(h)m jj = g f(a)m + hf V (a)g V f(a)m + o(h) ,}„ ‡!¤ ‰w –l y7l €7›2–†|Iq ¤„  ˜ l ›„‡ Yx y ˆ€ d0– ‡z | ˆ –}l‡V›y7y7€77€ll ª––}Dl€””€…¦zDÚ yž…¦¦…)‡7€ 88¦g|l¤– ††„„©0}–yr‡‡ „†‡V8–sx˜„€–}©l˜!›€l xl©l|˜rx„›D—uw ˜‡‡l™|l|l|xu y—y€sƒ„}–˜‡‘w| ylw|˜‰)„• ywq›l ‡y”|gl„ƒlg¦ v0j›!–a––}l€z‡|zt,|l|Q§¦f l(8yl©}–a||hx y)y7…–†yml¦…0l7€s‡€=}–¤„|‡Vxds h€hyxl –}uˆ ||ˆf¤€ yw©gwƒzV‡ eÛl|ew ©x8j l©l|wu x˜ €˜8–}l)„¤|z‡c| !Ûgf§¦˜ ƒ„−)l‡• w1 uxyg|w y€–†0e|…–))zl–©`Ûe‡|z€ˆŠ©|flfy7•ˆl©w y–}x|g˜h|8tvƒ– 08|l… ¥–e|lu‡fzV‡€l| xy€ˆ|qx –¿sy7l sd˜l˜| ©ll˜‰ll“|‰xYۉl7y‡Rfy‰Ûwg|g)…|‰§˜–| ˜l– –ƒ˜8 „› l ‡l| |x w l|–}}–˜¥x || u€fyl−¤–| ©1”y8l „ |‰‡l ƒ„ ||‡ „ x


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