<div id="pf1" class="pf w0 h0" data-page-no="1"><div class="pc pc1 w0 h0"><img class="bi x0 y0 w1 h1" alt src="https://files.passeidireto.com/b4885e16-a4d8-4670-ac9b-e7d7f04d2050/bg1.png" alt="Pré-visualização de imagem de arquivo"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">GEIVISON DOS SANTOS RIBEIR<span class="blank _0"></span>O</div><div class="t m0 x2 h3 y2 ff1 fs1 fc0 sc0 ls0 ws1">Linearidade em conjun<span class="blank _1"></span>tos de fun¸<span class="blank _2"></span>c˜<span class="blank _3"></span>oes</div><div class="t m0 x3 h3 y3 ff1 fs1 fc0 sc0 ls0 ws1">con<span class="blank _1"></span>t<span class="blank _4"></span>´<span class="blank _5"></span>ın<span class="blank _1"></span>uas que n˜<span class="blank _3"></span>ao s˜<span class="blank _3"></span>ao diferenci´<span class="blank _3"></span>av<span class="blank _1"></span>eis em</div><div class="t m0 x4 h3 y4 ff1 fs1 fc0 sc0 ls0 ws2">nenh<span class="blank _1"></span>um<span class="blank _6"> </span>p on<span class="blank _1"></span>to</div><div class="t m0 x5 h4 y5 ff1 fs0 fc0 sc0 ls0 ws0">UNIVERSID<span class="blank _0"></span>ADE FEDERAL DE UBERL<span class="blank _7"> </span><span class="v1">ˆ</span></div><div class="t m0 x6 h2 y5 ff1 fs0 fc0 sc0 ls0">ANDIA</div><div class="t m0 x7 h4 y6 ff1 fs0 fc0 sc0 ls0 ws3">F<span class="blank _8"></span>A<span class="blank _0"></span>CULD<span class="blank _0"></span>ADE<span class="blank _9"> </span>DE<span class="blank _9"> </span>MA<span class="blank _a"></span>TEM <span class="v1">´</span></div><div class="t m0 x8 h2 y6 ff1 fs0 fc0 sc0 ls0">ATICA</div><div class="t m0 x9 h2 y7 ff1 fs0 fc0 sc0 ls0">2019</div><div class="t m0 xa h2 y8 ff2 fs0 fc0 sc0 ls0">i</div></div><div class="pi" data-data="{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}"></div></div> <div id="pf2" class="pf w0 h0" data-page-no="2"><div class="pc pc2 w0 h0"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">GEIVISON DOS SANTOS RIBEIR<span class="blank _0"></span>O</div><div class="t m0 xb h3 y9 ff1 fs1 fc0 sc0 ls0 ws1">Linearidade em conjun<span class="blank _1"></span>tos de fun¸<span class="blank _2"></span>c˜<span class="blank _3"></span>oes</div><div class="t m0 x3 h3 ya ff1 fs1 fc0 sc0 ls0 ws1">con<span class="blank _1"></span>t<span class="blank _4"></span>´<span class="blank _5"></span>ın<span class="blank _1"></span>uas que n˜<span class="blank _3"></span>ao s˜<span class="blank _3"></span>ao diferenci´<span class="blank _3"></span>av<span class="blank _1"></span>eis em</div><div class="t m0 x4 h3 yb ff1 fs1 fc0 sc0 ls0 ws2">nenh<span class="blank _1"></span>um<span class="blank _6"> </span>p on<span class="blank _1"></span>to</div><div class="t m0 xc h2 yc ff1 fs0 fc0 sc0 ls0 ws5">Disserta¸<span class="blank _b"></span>c˜<span class="blank _c"></span>ao <span class="ff2 ws6">apresen<span class="blank _0"></span>tada ao Programa de P´<span class="blank _b"></span>os-</span></div><div class="t m0 xc h2 yd ff2 fs0 fc0 sc0 ls0 ws7">Gradua¸<span class="blank _d"></span>c˜<span class="blank _b"></span>ao em Matem´<span class="blank _e"></span>atica da Univ<span class="blank _0"></span>ersidade F<span class="blank _a"></span>ederal de</div><div class="t m0 xc h2 ye ff2 fs0 fc0 sc0 ls0 ws8">Ub<span class="blank _f"> </span>erlˆ<span class="blank _b"></span>andia, como parte dos requisitos para obten¸<span class="blank _d"></span>c˜<span class="blank _e"></span>ao do</div><div class="t m0 xc h4 yf ff2 fs0 fc0 sc0 ls0 ws9">t<span class="blank _8"></span>´<span class="blank _10"></span>ıtulo de <span class="ff1 ws0">MESTRE EM MA<span class="blank _a"></span>TEM<span class="blank _7"> </span><span class="v1">´</span></span></div><div class="t m0 xd h2 yf ff1 fs0 fc0 sc0 ls0">ATICA.</div><div class="t m0 xe h2 y10 ff1 fs0 fc0 sc0 ls0">´</div><div class="t m0 xc h2 y11 ff1 fs0 fc0 sc0 ls0 ws0">Area de Concen<span class="blank _0"></span>tra¸<span class="blank _b"></span>c˜<span class="blank _c"></span>ao:<span class="blank _11"> </span><span class="ff2 wsa">Matem´<span class="blank _b"></span>atica.</span></div><div class="t m0 xc h2 y12 ff1 fs0 fc0 sc0 ls0 ws0">Linha de P<span class="blank _0"></span>esquisa:<span class="blank _11"> </span><span class="ff2 wsb">An´<span class="blank _b"></span>alise F<span class="blank _1"></span>uncional.</span></div><div class="t m0 xc h2 y13 ff1 fs0 fc0 sc0 ls0 wsc">Orien<span class="blank _0"></span>tador: <span class="ff2 wsd">Prof.<span class="blank _11"> </span>Dr.<span class="blank _11"> </span>Vin<span class="blank _8"></span>´<span class="blank _10"></span>ıcius Vieira F´<span class="blank _b"></span>av<span class="blank _1"></span>aro.</span></div><div class="t m0 xf h4 y14 ff1 fs0 fc0 sc0 ls0 wse">UBERL <span class="v1">ˆ</span></div><div class="t m0 x10 h2 y14 ff1 fs0 fc0 sc0 ls0 ws0">ANDIA - MG</div><div class="t m0 x9 h2 y15 ff1 fs0 fc0 sc0 ls0">2019</div><div class="t m0 x11 h2 y8 ff2 fs0 fc0 sc0 ls0">ii</div></div><div class="pi" data-data="{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}"></div></div> <div id="pf3" class="pf w2 h5" data-page-no="3"><div class="pc pc3 w2 h5"><img fetchpriority="low" loading="lazy" class="bi x12 y16 w3 h6" alt src="https://files.passeidireto.com/b4885e16-a4d8-4670-ac9b-e7d7f04d2050/bg3.png" alt="Pré-visualização de imagem de arquivo"><div class="t m0 x13 h7 y17 ff3 fs2 fc0 sc0 ls0 ws4">Ribeiro, Geivison dos Santos, 1988-<span class="blank _12"></span>R484</div><div class="t m0 x14 h8 y18 ff3 fs2 fc0 sc0 ls0 wsf">2019 <span class="ws4 v2">Linearidade em conjuntos de funções contínuas que não são</span></div><div class="t m0 x13 h7 y19 ff3 fs2 fc0 sc0 ls0 ws4">diferenciáveis em nenhum ponto [recurso eletrônico] / Geivison</div><div class="t m0 x13 h7 y1a ff3 fs2 fc0 sc0 ls0 ws4">dos Santos Ribeiro. - 2019.</div><div class="t m0 x15 h7 y1b ff3 fs2 fc0 sc0 ls0 ws4">Orientador: Vinícius Vieira Fávaro.</div><div class="t m0 x15 h7 y1c ff3 fs2 fc0 sc0 ls0 ws4">Dissertação (Mestrado) - Universidade Federal de Uberlândia,</div><div class="t m0 x13 h7 y1d ff3 fs2 fc0 sc0 ls0 ws4">Pós-graduação em Matemática.</div><div class="t m0 x15 h7 y1e ff3 fs2 fc0 sc0 ls0 ws4">Modo de acesso: Internet.</div><div class="t m0 x16 h7 y1f ff3 fs2 fc0 sc0 ls0 ws4">CDU: 51</div><div class="t m0 x15 h7 y20 ff3 fs2 fc0 sc0 ls0 ws4">1. Matemática. I. Fávaro, Vinícius Vieira, 81 -, (Orient.). II.</div><div class="t m0 x13 h7 y21 ff3 fs2 fc0 sc0 ls0 ws4">Universidade Federal de Uberlândia. Pós-graduação em</div><div class="t m0 x13 h7 y22 ff3 fs2 fc0 sc0 ls0 ws4">Matemática. III. Título.</div><div class="t m0 x15 h7 y23 ff3 fs2 fc0 sc0 ls0 ws4">Disponível em: http://doi.org/10.14393/ufu.di.2019.2471</div><div class="t m0 x15 h7 y24 ff3 fs2 fc0 sc0 ls0 ws4">Inclui bibliografia.</div><div class="t m0 x17 h7 y25 ff3 fs2 fc0 sc0 ls0 ws4">Ficha Catalográfica Online do Sistema de Bibliotecas da UFU</div><div class="t m0 x18 h7 y26 ff3 fs2 fc0 sc0 ls0 ws4">com dados informados pelo(a) próprio(a) autor(a).</div><div class="t m0 x1 h7 y27 ff3 fs2 fc0 sc0 ls0 ws4">Bibliotecários responsáveis pela estrutura de acordo com o AACR2:</div><div class="t m0 x19 h7 y28 ff3 fs2 fc0 sc0 ls0 ws4">Gizele Cristine Nunes do Couto - CRB6/2091</div><div class="t m0 x1a h7 y29 ff3 fs2 fc0 sc0 ls0 ws4">Nelson Marcos Ferreira - CRB6/3074</div></div><div class="pi" data-data="{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}"></div></div> <div id="pf4" class="pf w2 h5" data-page-no="4"><div class="pc pc4 w2 h5"><img fetchpriority="low" loading="lazy" class="bi x1b y2a w4 h9" alt src="https://files.passeidireto.com/b4885e16-a4d8-4670-ac9b-e7d7f04d2050/bg4.png" alt="Pré-visualização de imagem de arquivo"><div class="t m0 x1c ha y2b ff4 fs3 fc0 sc0 ls0 ws4">Scanned by CamScanner</div></div><div class="pi" data-data="{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}"></div></div> <div id="pf5" class="pf w0 h0" data-page-no="5"><div class="pc pc5 w0 h0"><div class="t m0 x1d h3 y2c ff1 fs1 fc0 sc0 ls0 ws10">Dedicat´<span class="blank _3"></span>oria</div><div class="t m0 x1e h2 y2d ff2 fs0 fc0 sc0 ls0 wsd">A minha m˜<span class="blank _b"></span>ae Mirab<span class="blank _f"> </span>el Gonzaga e a minha esp<span class="blank _f"> </span>osa Andr´<span class="blank _e"></span>ea T<span class="blank _a"></span>eles com muito amor.</div><div class="t m0 x11 h2 y8 ff2 fs0 fc0 sc0 ls0">v</div></div><div class="pi" data-data="{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}"></div></div> <div id="pf6" class="pf w0 h0" data-page-no="6"><div class="pc pc6 w0 h0"><div class="t m0 x1f h3 y2e ff1 fs1 fc0 sc0 ls0 ws10">Agradecimen<span class="blank _1"></span>tos</div><div class="t m0 x1e h2 y2f ff2 fs0 fc0 sc0 ls0 wsb">Agrade¸<span class="blank _d"></span>co primeiramen<span class="blank _0"></span>te a Deus p<span class="blank _f"> </span>or sua not´<span class="blank _b"></span>avel benevol<span class="blank _1"></span>ˆ<span class="blank _d"></span>encia para comigo.<span class="blank _11"> </span>P<span class="blank _1"></span>or to<span class="blank _f"> </span>do</div><div class="t m0 x20 h2 y30 ff2 fs0 fc0 sc0 ls0 ws11">amor e cuidado atribu<span class="blank _8"></span>´<span class="blank _10"></span>ıdo a mim dian<span class="blank _0"></span>te n˜<span class="blank _b"></span>ao somente dos momen<span class="blank _1"></span>tos f´<span class="blank _b"></span>aceis,<span class="blank _9"> </span>mas tam<span class="blank _0"></span>b´<span class="blank _b"></span>em</div><div class="t m0 x20 h2 y31 ff2 fs0 fc0 sc0 ls0 ws12">dos momen<span class="blank _0"></span>tos dif<span class="blank _8"></span>´<span class="blank _10"></span>ıceis.<span class="blank _13"> </span>Agrade¸<span class="blank _d"></span>co aos meus pais, Mirab<span class="blank _f"> </span>el Gonzaga dos San<span class="blank _0"></span>tos e Gen´<span class="blank _e"></span>esio</div><div class="t m0 x20 h2 y32 ff2 fs0 fc0 sc0 ls0 ws13">Rib<span class="blank _f"> </span>eiro (em esp<span class="blank _f"> </span>ecial, a minha m˜<span class="blank _e"></span>ae, p<span class="blank _f"> </span>or sempre torcer p<span class="blank _f"> </span>or mim),<span class="blank _14"> </span>as minhas irm˜<span class="blank _b"></span>as Cl´<span class="blank _e"></span>audia</div><div class="t m0 x20 h2 y33 ff2 fs0 fc0 sc0 ls0 ws14">e Cristina,<span class="blank _11"> </span>aos meus amigos Jo<span class="blank _f"> </span>ed Muritiba,<span class="blank _15"> </span>Fidel Eduardo e Rafael Reis,<span class="blank _11"> </span>p<span class="blank _f"> </span>elo grande</div><div class="t m0 x20 h2 y34 ff2 fs0 fc0 sc0 ls0 ws15">carinho que creditaram a mim, ao meu amigo e irm˜<span class="blank _b"></span>ao Ismael, p<span class="blank _f"> </span>or sua imensa paciˆ<span class="blank _e"></span>encia,</div><div class="t m0 x20 h2 y35 ff2 fs0 fc0 sc0 ls0 ws16">amizade e simplicidade, ao meu companheiro de Cristianismo Pr.<span class="blank _11"> </span>Luiz, p<span class="blank _f"> </span>or seu cotidiano</div><div class="t m0 x20 h2 y36 ff2 fs0 fc0 sc0 ls0 ws17">exemplo de amor para com o rebanho de Cristo, a minha amada esp<span class="blank _f"> </span>osa que esteve comigo</div><div class="t m0 x20 h2 y37 ff2 fs0 fc0 sc0 ls0 ws18">duran<span class="blank _0"></span>te to<span class="blank _f"> </span>dos esses anos e ao meu filho Davi, por to<span class="blank _f"> </span>das as vezes que c<span class="blank _1"></span>heguei cansado ou</div><div class="t m0 x20 h2 y38 ff2 fs0 fc0 sc0 ls0 ws19">triste e me acolheram com to<span class="blank _f"> </span>do o amor e carinho ap<span class="blank _f"> </span>oiando-me nos meus planos e n<span class="blank _0"></span>unca me</div><div class="t m0 x20 h2 y39 ff2 fs0 fc0 sc0 ls0 ws1a">deixando desistir p<span class="blank _f"> </span>elas dificuldades.<span class="blank _16"> </span>Obrigada p<span class="blank _f"> </span>ela luta e dedica¸<span class="blank _d"></span>c˜<span class="blank _b"></span>ao durante toda a minha</div><div class="t m0 x20 h2 y3a ff2 fs0 fc0 sc0 ls0 ws1b">vida, em esp<span class="blank _f"> </span>ecial nesse p<span class="blank _f"> </span>er<span class="blank _a"></span>´<span class="blank _17"></span>ıo<span class="blank _f"> </span>do t˜<span class="blank _b"></span>ao exaustivo...<span class="blank _16"> </span>V<span class="blank _a"></span>o<span class="blank _f"> </span>cˆ<span class="blank _b"></span>es s˜<span class="blank _e"></span>ao tudo para mim e tudo o que eu</div><div class="t m0 x20 h2 y3b ff2 fs0 fc0 sc0 ls0 ws1c">sou<span class="blank _18"> </span>´<span class="blank _e"></span>e p<span class="blank _f"> </span>or vocˆ<span class="blank _e"></span>es!<span class="blank _16"> </span>Agrade¸<span class="blank _d"></span>co a to<span class="blank _f"> </span>dos os meus tios, primos,<span class="blank _14"> </span>em esp<span class="blank _f"> </span>ecial `<span class="blank _b"></span>a minha tia Ducineia,</div><div class="t m0 x20 h2 y3c ff2 fs0 fc0 sc0 ls0 ws1d">p<span class="blank _f"> </span>or sempre me estender a m˜<span class="blank _b"></span>ao amiga.<span class="blank _11"> </span>Agrade¸<span class="blank _d"></span>co ao meu nobre amigo Diego Alv<span class="blank _0"></span>es p<span class="blank _f"> </span>or me</div><div class="t m0 x20 h2 y3d ff2 fs0 fc0 sc0 ls0 ws1e">ap<span class="blank _f"> </span>oiar nos momen<span class="blank _0"></span>tos mais dif<span class="blank _8"></span>´<span class="blank _10"></span>ıceis com rela¸<span class="blank _d"></span>c˜<span class="blank _b"></span>ao aos cursos de matem´<span class="blank _b"></span>atica (gradua¸<span class="blank _d"></span>c˜<span class="blank _e"></span>ao e</div><div class="t m0 x20 h2 y3e ff2 fs0 fc0 sc0 ls0 ws1f">mestrado) com seu grande cora¸<span class="blank _d"></span>c˜<span class="blank _b"></span>ao.<span class="blank _11"> </span>P<span class="blank _0"></span>or ser verdadeiro, leal e pacien<span class="blank _1"></span>te.<span class="blank _11"> </span>Por acreditar que</div><div class="t m0 x20 h2 y3f ff2 fs0 fc0 sc0 ls0 ws20">eu p<span class="blank _f"> </span>o<span class="blank _f"> </span>deria c<span class="blank _0"></span>hegar a concluir o mestrado.<span class="blank _19"> </span>Agrade¸<span class="blank _d"></span>co-o p<span class="blank _f"> </span>or to<span class="blank _f"> </span>da a<span class="blank _1a"> </span>juda de custo, p<span class="blank _f"> </span>ois sem</div><div class="t m0 x20 h2 y40 ff2 fs0 fc0 sc0 ls0 ws19">isso, n˜<span class="blank _e"></span>ao sei se teria condi¸<span class="blank _d"></span>c˜<span class="blank _b"></span>oes de chegar a estudar em Uberlˆ<span class="blank _e"></span>andia...<span class="blank _16"> </span>Agrade¸<span class="blank _d"></span>co a Cap<span class="blank _f"> </span>es p<span class="blank _f"> </span>elo</div><div class="t m0 x20 h2 y41 ff2 fs0 fc0 sc0 ls0 ws21">ap<span class="blank _f"> </span>oio financeiro, `<span class="blank _b"></span>a Universidade F<span class="blank _a"></span>ederal de Ub<span class="blank _f"> </span>erlˆ<span class="blank _b"></span>andia e `<span class="blank _b"></span>a F<span class="blank _1"></span>aculdade de Matem´<span class="blank _b"></span>atica p<span class="blank _f"> </span>or</div><div class="t m0 x20 h2 y42 ff2 fs0 fc0 sc0 ls0 ws22">minha forma¸<span class="blank _d"></span>c˜<span class="blank _b"></span>ao acadˆ<span class="blank _b"></span>emica,<span class="blank _1b"> </span>e p<span class="blank _f"> </span>or me apresen<span class="blank _0"></span>tar os melhores professores que eu p<span class="blank _f"> </span>o<span class="blank _f"> </span>deria ter</div><div class="t m0 x20 h2 y43 ff2 fs0 fc0 sc0 ls0 ws23">tido duran<span class="blank _0"></span>te o curso de mestrado.<span class="blank _11"> </span>Agrade¸<span class="blank _d"></span>co aos amigos que fiz durante todos esses anos</div><div class="t m0 x20 h2 y44 ff2 fs0 fc0 sc0 ls0 ws24">de estudo no mestrado, em esp<span class="blank _1a"> </span>ecial ao meu amigo Geo<span class="blank _1"></span>v<span class="blank _1"></span>any<span class="blank _a"></span>,<span class="blank _16"> </span>p<span class="blank _f"> </span>ela sua imensa amizade,</div><div class="t m0 x20 h2 y45 ff2 fs0 fc0 sc0 ls0 ws25">companheirismo e simplicidade.<span class="blank _1c"> </span>Agrade¸<span class="blank _d"></span>co a cada professor que tiv<span class="blank _0"></span>e con<span class="blank _0"></span>tato p<span class="blank _f"> </span>or to<span class="blank _f"> </span>da</div><div class="t m0 x20 h2 y46 ff2 fs0 fc0 sc0 ls0 ws26">dedica¸<span class="blank _d"></span>c˜<span class="blank _b"></span>ao e ensinamento.<span class="blank _1d"> </span>Em esp<span class="blank _f"> </span>ecial,<span class="blank _13"> </span>ao professor Alonso pelos momentos em que</div><div class="t m0 x20 h2 y47 ff2 fs0 fc0 sc0 ls0 ws1e">estiv<span class="blank _0"></span>emos a tro<span class="blank _f"> </span>car ideias referen<span class="blank _0"></span>tes ao amor de Deus.<span class="blank _6"> </span>N˜<span class="blank _b"></span>ao p<span class="blank _f"> </span>o<span class="blank _f"> </span>deria deixar de ser grato</div><div class="t m0 x20 h2 y48 ff2 fs0 fc0 sc0 ls0 ws27">p<span class="blank _f"> </span>ela vida do professor Geraldo Botelho, um dos melhores professores que j´<span class="blank _b"></span>a tive a honra</div><div class="t m0 x20 h2 y49 ff2 fs0 fc0 sc0 ls0 wsd">de conhecer.</div><div class="t m0 x21 h2 y8 ff2 fs0 fc0 sc0 ls0">vi</div></div><div class="pi" data-data="{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}"></div></div> <div id="pf7" class="pf w0 h0" data-page-no="7"><div class="pc pc7 w0 h0"><div class="t m0 x1e h2 y1 ff2 fs0 fc0 sc0 ls0 ws28">Agrade¸<span class="blank _d"></span>co demais ao professor Vin<span class="blank _8"></span>´<span class="blank _17"></span>ıcius Vieira F´<span class="blank _e"></span>av<span class="blank _1"></span>aro (meu orientador) pela confian¸<span class="blank _d"></span>ca</div><div class="t m0 x20 h2 y4a ff2 fs0 fc0 sc0 ls0 ws29">n˜<span class="blank _b"></span>ao somente em mim, mas em meu trabalho.<span class="blank _1e"> </span>P<span class="blank _0"></span>or to<span class="blank _f"> </span>do o ap<span class="blank _f"> </span>oio que me passou,<span class="blank _15"> </span>p<span class="blank _f"> </span>ela</div><div class="t m0 x20 h2 y4b ff2 fs0 fc0 sc0 ls0 ws2a">amizade,<span class="blank _15"> </span>p<span class="blank _f"> </span>elas con<span class="blank _0"></span>ve<span class="blank _0"></span>rsas,<span class="blank _15"> </span>p<span class="blank _f"> </span>elos conselhos e incentiv<span class="blank _1"></span>os.<span class="blank _1c"> </span>Ao programa de Mestrado em</div><div class="t m0 x20 h2 y4c ff2 fs0 fc0 sc0 ls0 ws2b">Matem´<span class="blank _b"></span>atica da UFU, p<span class="blank _f"> </span>elo privil´<span class="blank _b"></span>egio e a op<span class="blank _f"> </span>ortunidade de fazer o curso.<span class="blank _16"> </span>Aos co<span class="blank _f"> </span>ordenadores</div><div class="t m0 x20 h2 y4d ff2 fs0 fc0 sc0 ls0 ws2c">Dr.<span class="blank _13"> </span>Thiago Aparecido Catalan, e a Dra.<span class="blank _13"> </span>Rosana Sueli da Motta Jafelice (em especial a</div><div class="t m0 x20 h2 y4e ff2 fs0 fc0 sc0 ls0 ws2d">Dra.<span class="blank _1f"> </span>Rosana Sueli da Motta Jafelice p<span class="blank _f"> </span>ela sua garra no que diz resp<span class="blank _f"> </span>eito ao ouvir cada</div><div class="t m0 x20 h2 y4f ff2 fs0 fc0 sc0 ls0 ws2e">aluno com o cora¸<span class="blank _d"></span>c˜<span class="blank _b"></span>ao) p<span class="blank _f"> </span>or serem t˜<span class="blank _e"></span>ao prestativ<span class="blank _0"></span>os e atenciosos sempre e p<span class="blank _f"> </span>or fim ao Dr.</div><div class="t m0 x20 h2 y50 ff2 fs0 fc0 sc0 ls0 ws2c">Wilb<span class="blank _f"> </span>ercla<span class="blank _0"></span>y Gon¸<span class="blank _d"></span>calv<span class="blank _0"></span>es de Melo, p<span class="blank _f"> </span>or ser quem foi durante a minha gradua¸<span class="blank _d"></span>c˜<span class="blank _b"></span>ao e p<span class="blank _f"> </span>or tudo</div><div class="t m0 x20 h2 y51 ff2 fs0 fc0 sc0 ls0 ws2f">aquilo que me ensinou relativ<span class="blank _0"></span>o a vida e principalmen<span class="blank _0"></span>te no que diz resp<span class="blank _f"> </span>eito a matem´<span class="blank _b"></span>atica.</div><div class="t m0 x22 h2 y8 ff2 fs0 fc0 sc0 ls0">vii</div></div><div class="pi" data-data="{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}"></div></div> <div id="pf8" class="pf w0 h0" data-page-no="8"><div class="pc pc8 w0 h0"><div class="t m0 x20 h2 y1 ff2 fs0 fc0 sc0 ls0 ws30">RIBEIR<span class="blank _0"></span>O. <span class="ff5 ws20">Line<span class="blank _1"></span>aridade em c<span class="blank _1"></span>onjuntos de fun¸<span class="blank _d"></span>c˜<span class="blank _b"></span>oes c<span class="blank _1"></span>ont<span class="blank _a"></span>´<span class="blank _20"></span>ınuas que n˜<span class="blank _b"></span>ao s˜<span class="blank _b"></span>ao difer<span class="blank _1"></span>enci´<span class="blank _b"></span>aveis em</span></div><div class="t m0 x20 h2 y4a ff5 fs0 fc0 sc0 ls0 ws31">nenhum p<span class="blank _1"></span>onto<span class="blank _14"> </span><span class="ff2 ws32">2019.<span class="blank _16"> </span>- 99p.<span class="blank _16"> </span>Disserta¸<span class="blank _d"></span>c˜<span class="blank _b"></span>ao de Mestrado,<span class="blank _1b"> </span>Univ<span class="blank _0"></span>ersidade F<span class="blank _1"></span>ederal de Uberlˆ<span class="blank _b"></span>andia,</span></div><div class="t m0 x20 h2 y4b ff2 fs0 fc0 sc0 ls0 ws33">Ub erlˆ<span class="blank _b"></span>andia-MG.</div><div class="t m0 x23 h2 y52 ff1 fs0 fc0 sc0 ls0">Resumo</div><div class="t m0 x20 h2 y53 ff2 fs0 fc0 sc0 ls0 ws34">Neste trabalho,<span class="blank _21"> </span>fazemos um estudo detalhado sobre fun¸<span class="blank _d"></span>c˜<span class="blank _b"></span>oes cont<span class="blank _8"></span>´<span class="blank _17"></span>ın<span class="blank _0"></span>uas que n˜<span class="blank _b"></span>ao s˜<span class="blank _e"></span>ao</div><div class="t m0 x20 h2 y54 ff2 fs0 fc0 sc0 ls0 ws35">diferenci´<span class="blank _b"></span>aveis em nenh<span class="blank _1"></span>um p<span class="blank _f"> </span>onto.<span class="blank _22"> </span>Come¸<span class="blank _d"></span>camos construindo um exemplo de tal fun¸<span class="blank _d"></span>c˜<span class="blank _b"></span>ao</div><div class="t m0 x20 hb y55 ff2 fs0 fc0 sc0 ls0 ws36">devido a v<span class="blank _1"></span>an der W<span class="blank _a"></span>aerden e,<span class="blank _19"> </span>em seguida,<span class="blank _13"> </span>pro<span class="blank _1"></span>v<span class="blank _1"></span>amos que o conjunto <span class="ff6 ws37">N<span class="blank _7"> </span>D </span><span class="wsa">[0<span class="ff7 ls1">,</span></span>1] de tais</div><div class="t m0 x20 hb y56 ff2 fs0 fc0 sc0 ls0 ws38">fun¸<span class="blank _d"></span>c˜<span class="blank _b"></span>oes ´<span class="blank _b"></span>e largo no sentido da categoria Baire em <span class="ff6 ls2">C</span><span class="wsa">[0<span class="ff7 ls1">,</span><span class="ws39">1].<span class="blank _16"> </span>Prov<span class="blank _a"></span>amos tamb<span class="blank _1"></span>´<span class="blank _d"></span>em que <span class="ff6 ws37">N<span class="blank _7"> </span>D </span><span class="wsa">[0<span class="ff7 ls1">,</span>1]</span></span></span></div><div class="t m0 x20 h2 y57 ff2 fs0 fc0 sc0 ls0 ws3a">´<span class="blank _e"></span>e espa¸<span class="blank _d"></span>c´<span class="blank _e"></span>av<span class="blank _0"></span>el,<span class="blank _19"> </span>mais ainda que os espa¸<span class="blank _d"></span>cos de Banach separ´<span class="blank _b"></span>av<span class="blank _0"></span>eis p<span class="blank _f"> </span>o<span class="blank _f"> </span>dem ser vistos como</div><div class="t m0 x20 hb y58 ff2 fs0 fc0 sc0 ls0 ws33">sub espa¸<span class="blank _d"></span>cos<span class="blank _23"> </span>de<span class="blank _23"> </span><span class="ff6 ws37">N<span class="blank _7"> </span>D </span><span class="wsa">[0<span class="ff7 ls1">,</span><span class="ws3b">1] <span class="ff6 ws3c">∪ {</span></span>0<span class="ff6 ls3">}</span><span class="ws3d">.<span class="blank _15"> </span>Finalmente, pro<span class="blank _1"></span>v<span class="blank _1"></span>amos que o conjunto das fun¸<span class="blank _d"></span>c˜<span class="blank _b"></span>oes H¨<span class="blank _b"></span>older</span></span></div><div class="t m0 x20 hb y59 ff2 fs0 fc0 sc0 ls0 ws8">em nenh<span class="blank _0"></span>um lugar<span class="blank _1b"> </span>´<span class="blank _e"></span>e denso-algebr´<span class="blank _e"></span>av<span class="blank _0"></span>el e, em particular,<span class="blank _14"> </span>obtemos que <span class="ff6 ws37">N<span class="blank _7"> </span>D </span><span class="wsa">[0<span class="ff7 ls1">,</span><span class="ws3e">1] ´<span class="blank _d"></span>e algebr´<span class="blank _e"></span>av<span class="blank _0"></span>el.</span></span></div><div class="t m0 x20 h2 y5a ff5 fs0 fc0 sc0 ls0 ws3f">Palavr<span class="blank _1"></span>as-chave <span class="ff2 ws32">:<span class="blank _16"> </span>Espa¸<span class="blank _d"></span>co de fun¸<span class="blank _d"></span>c˜<span class="blank _b"></span>oes con<span class="blank _0"></span>t<span class="blank _8"></span>´<span class="blank _10"></span>ın<span class="blank _0"></span>uas,<span class="blank _1b"> </span>fun¸<span class="blank _d"></span>c˜<span class="blank _b"></span>oes con<span class="blank _0"></span>t<span class="blank _a"></span>´<span class="blank _20"></span>ınuas que n˜<span class="blank _b"></span>ao s˜<span class="blank _b"></span>ao diferenci´<span class="blank _e"></span>av<span class="blank _0"></span>eis</span></div><div class="t m0 x20 h2 y5b ff2 fs0 fc0 sc0 ls0 wsd">em nenh<span class="blank _0"></span>um p<span class="blank _f"> </span>on<span class="blank _0"></span>to, espa¸<span class="blank _d"></span>cabilidade, algebralidade.</div><div class="t m0 x24 h2 y8 ff2 fs0 fc0 sc0 ls0">viii</div></div><div class="pi" data-data="{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}"></div></div> <div id="pf9" class="pf w0 h0" data-page-no="9"><div class="pc pc9 w0 h0"><div class="t m0 x20 h2 y1 ff2 fs0 fc0 sc0 ls0 ws40">RIBEIR<span class="blank _0"></span>O, <span class="ff5 ws41">Line<span class="blank _1"></span>arity in sets of nowher<span class="blank _1"></span>e differ<span class="blank _1"></span>entiable c<span class="blank _1"></span>ontinuous<span class="blank _9"> </span>functions.<span class="blank _19"> </span><span class="ff2 ws42">2019.<span class="blank _19"> </span>- 99p.</span></span></div><div class="t m0 x20 h2 y4a ff2 fs0 fc0 sc0 ls0 wsd">M. Sc.<span class="blank _11"> </span>Dissertation, F<span class="blank _a"></span>ederal Universit<span class="blank _1"></span>y of Ub<span class="blank _f"> </span>erlˆ<span class="blank _e"></span>andia, Ub<span class="blank _f"> </span>erlˆ<span class="blank _b"></span>andia-MG.</div><div class="t m0 x25 h2 y5c ff1 fs0 fc0 sc0 ls0">Abstract</div><div class="t m0 x20 h2 y5d ff2 fs0 fc0 sc0 ls0 ws43">In this w<span class="blank _0"></span>ork w<span class="blank _0"></span>e study in detail the set of nowhere differen<span class="blank _1"></span>tiable contin<span class="blank _1"></span>uous functions.<span class="blank _16"> </span>W<span class="blank _1"></span>e</div><div class="t m0 x20 h2 y53 ff2 fs0 fc0 sc0 ls0 ws44">start constructing an example of suc<span class="blank _0"></span>h function due to v<span class="blank _1"></span>an der W<span class="blank _a"></span>aeden<span class="blank _23"> </span>and w<span class="blank _1"></span>e prov<span class="blank _1"></span>e that</div><div class="t m0 x20 hb y54 ff2 fs0 fc0 sc0 ls0 ws45">the set <span class="ff6 ws37">N<span class="blank _7"> </span>D </span><span class="wsa">[0<span class="ff7 ls1">,</span></span>1] of suc<span class="blank _0"></span>h functions is large in <span class="ff6 ls2">C</span><span class="wsa">[0<span class="ff7 ls1">,</span></span>1], in the sense of Baire category<span class="blank _1"></span>.<span class="blank _24"> </span>W<span class="blank _a"></span>e</div><div class="t m0 x20 hb y55 ff2 fs0 fc0 sc0 ls0 ws46">also pro<span class="blank _0"></span>v<span class="blank _0"></span>e that <span class="ff6 ws37">N<span class="blank _7"> </span>D </span><span class="wsa">[0<span class="ff7 ls1">,</span></span>1] is spaceable and, moreov<span class="blank _1"></span>er, the separable Banach spaces can be</div><div class="t m0 x20 hb y56 ff2 fs0 fc0 sc0 ls0 ws47">seen as subspaces of <span class="ff6 ls2">C</span><span class="wsa">[0<span class="ff7 ls4">,</span></span>1].<span class="blank _16"> </span>Finally<span class="blank _1"></span>, w<span class="blank _1"></span>e<span class="blank _23"> </span>pro<span class="blank _0"></span>v<span class="blank _0"></span>e that the set of nowhere H¨<span class="blank _b"></span>older functions is</div><div class="t m0 x20 hb y57 ff2 fs0 fc0 sc0 ls0 wsd">dense-algebrable and in particular w<span class="blank _0"></span>e obtain that <span class="ff6 ws37">N<span class="blank _7"> </span>D </span><span class="wsa">[0<span class="ff7 ls1">,</span></span>1] is algebrable.</div><div class="t m0 x20 h2 y5e ff5 fs0 fc0 sc0 ls0 ws48">Keywor<span class="blank _1"></span>ds <span class="ff2 ws2a">:<span class="blank _6"> </span>Space of con<span class="blank _0"></span>tinuous functions, nowhere differen<span class="blank _1"></span>tiable contin<span class="blank _1"></span>uous functions,</span></div><div class="t m0 x20 h2 y5f ff2 fs0 fc0 sc0 ls0 wsd">spaceabilit<span class="blank _0"></span>y<span class="blank _a"></span>, algebrability<span class="blank _a"></span>.</div><div class="t m0 x21 h2 y8 ff2 fs0 fc0 sc0 ls0">ix</div></div><div class="pi" data-data="{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}"></div></div> <div id="pfa" class="pf w0 h0" data-page-no="a"><div class="pc pca w0 h0"><img fetchpriority="low" loading="lazy" class="bi x20 y60 w5 hc" alt src="https://files.passeidireto.com/b4885e16-a4d8-4670-ac9b-e7d7f04d2050/bga.png" alt="Pré-visualização de imagem de arquivo"><div class="t m0 x26 hd y61 ff8 fs4 fc0 sc0 ls0 ws49">LIST<span class="blank _25"></span>A DE S</div><div class="t m0 x27 hd y62 ff8 fs4 fc0 sc0 ls0">´</div><div class="t m0 x28 hd y61 ff8 fs4 fc0 sc0 ls0">IMBOLOS</div><div class="t m0 x29 hb y63 ff6 fs0 fc0 sc0 ls5">∅<span class="ff2 ls0 wsd">Conjun<span class="blank _0"></span>to v<span class="blank _1"></span>azio</span></div><div class="t m0 x29 he y64 ff9 fs0 fc0 sc0 ls6">N<span class="ff2 ls0 wsd">Conjun<span class="blank _0"></span>to dos n<span class="blank _f"> </span>´<span class="blank _b"></span>umeros naturais</span></div><div class="t m0 x29 he y65 ff9 fs0 fc0 sc0 ls6">R<span class="ff2 ls0 wsd">Conjun<span class="blank _0"></span>to dos n<span class="blank _f"> </span>´<span class="blank _b"></span>umeros reais</span></div><div class="t m0 x29 hf y66 ff7 fs0 fc0 sc0 ls0 ws4a">`<span class="ffa fs5 ls7 v3">1</span><span class="ff2 wsd">Conjun<span class="blank _0"></span>to das sequ<span class="blank _0"></span>ˆ<span class="blank _d"></span>encias reais <span class="ff7 ls8">x</span><span class="ws4b">= (<span class="ff7 ls9">x<span class="ffb fs5 lsa v3">n</span></span><span class="wsa">)<span class="ffc fs5 v4">∞</span></span></span></span></div><div class="t m0 x2a h10 y67 ffb fs5 fc0 sc0 ls0 ws4c">n<span class="ffa ws4d">=1 <span class="ff2 fs0 wsd v1">cuja s<span class="blank _0"></span>´<span class="blank _26"></span>erie</span></span></div><div class="t m0 x2b h11 y68 ffc fs5 fc0 sc0 ls0">∞</div><div class="t m0 x2c h12 y69 ffd fs0 fc0 sc0 ls0">X</div><div class="t m0 x2c h11 y6a ffb fs5 fc0 sc0 ls0 ws4c">n<span class="ffa">=1</span></div><div class="t m0 x2d hb y66 ff6 fs0 fc0 sc0 ls0 ws4e">|<span class="ff7 ls9">x<span class="ffb fs5 lsa v3">n</span></span><span class="lsb">|</span><span class="ff2 wsa">con<span class="blank _0"></span>v<span class="blank _0"></span>erge</span></div><div class="t m0 x29 h2 y6b ff7 fs0 fc0 sc0 lsc">E<span class="ff2 ls0 wsd">Espa¸<span class="blank _d"></span>co v<span class="blank _0"></span>etorial real</span></div><div class="t m0 x29 h2 y6c ff7 fs0 fc0 sc0 ls0 ws4a">span<span class="ff2 wsa">(</span><span class="lsd">A</span><span class="ff2 wsd">)<span class="blank _27"> </span>Sub<span class="blank _f"> </span>espa¸<span class="blank _d"></span>co linear gerado p<span class="blank _f"> </span>elo conjun<span class="blank _0"></span>to <span class="ff7">A</span></span></div><div class="t m0 x29 h2 y6d ff7 fs0 fc0 sc0 lse">A<span class="ff2 ls0 wsd">F<span class="blank _a"></span>echo do conjun<span class="blank _1"></span>to <span class="ff7">A</span></span></div><div class="t m0 x29 h2 y6e ff7 fs0 fc0 sc0 ls0 ws4a">B<span class="ffb fs5 lsf v3">r</span><span class="ff2 ls10">(</span><span class="ls11">f</span><span class="ff2 wsd">)<span class="blank _28"> </span>Bola ab<span class="blank _f"> </span>erta de cen<span class="blank _0"></span>tro <span class="ff7 ls12">f</span>e raio <span class="ff7 ws4f">r<span class="blank _14"> </span>> </span>0</span></div><div class="t m0 x29 he y6f ff6 fs0 fc0 sc0 ls2">C<span class="ff2 ls0 wsa">[0<span class="ff7 ls1">,</span><span class="wsd">1]<span class="blank _29"> </span>Espa¸<span class="blank _d"></span>co v<span class="blank _0"></span>etorial de to<span class="blank _f"> </span>das as fun¸<span class="blank _d"></span>c˜<span class="blank _b"></span>oes cont<span class="blank _8"></span>´<span class="blank _17"></span>ınuas de [0<span class="ff7 ls1">,</span>1] em <span class="ff9">R</span></span></span></div><div class="t m0 x29 he y70 ff6 fs0 fc0 sc0 ls0 ws37">N<span class="blank _7"> </span>D <span class="ff2 wsa">[0<span class="ff7 ls1">,</span><span class="wsb">1]<span class="blank _2a"> </span>Conjun<span class="blank _0"></span>to de to<span class="blank _f"> </span>das as fun¸<span class="blank _d"></span>c˜<span class="blank _b"></span>oes cont<span class="blank _8"></span>´<span class="blank _17"></span>ın<span class="blank _0"></span>uas de [0<span class="ff7 ls1">,</span><span class="wsd">1] em <span class="ff9 ls13">R</span>que</span></span></span></div><div class="t m0 x2e h2 y71 ff2 fs0 fc0 sc0 ls0 wsd">s˜<span class="blank _b"></span>ao n˜<span class="blank _e"></span>ao diferenci´<span class="blank _b"></span>aveis em nenh<span class="blank _1"></span>um p<span class="blank _f"> </span>onto</div><div class="t m0 x29 he y72 ff6 fs0 fc0 sc0 ls0 ws37">N<span class="blank _7"> </span>D <span class="ffc fs5 ls14 v3">±</span><span class="ff2 ls10">(</span><span class="ff7 ws4a">A<span class="ff2 wsd">)<span class="blank _2b"> </span>Conjun<span class="blank _0"></span>to de to<span class="blank _f"> </span>das as fun¸<span class="blank _d"></span>c˜<span class="blank _b"></span>oes cont<span class="blank _8"></span>´<span class="blank _17"></span>ınuas de <span class="ff7 ls15">A</span><span class="ws50">em <span class="ff9 ls16">R</span>que</span></span></span></div><div class="t m0 x2e h2 y73 ff2 fs0 fc0 sc0 ls0 wsd">n˜<span class="blank _b"></span>ao p<span class="blank _f"> </span>ossuem deriv<span class="blank _1"></span>ada finita a esquerda nem tamp<span class="blank _f"> </span>ouco a direita de cada p<span class="blank _f"> </span>onto</div><div class="t m0 x29 h13 y74 ff6 fs0 fc0 sc0 ls17">A<span class="ff2 ls0 v1">´</span></div><div class="t m0 x2e h2 y74 ff2 fs0 fc0 sc0 ls0 wsd">Algebra v<span class="blank _0"></span>etorial</div><div class="t m0 x29 h13 y75 ff6 fs0 fc0 sc0 ls0 ws4e">A<span class="ff2 wsa">(<span class="ff7 ls18">u</span><span class="ls19">)</span><span class="v1">´</span></span></div><div class="t m0 x2e h2 y75 ff2 fs0 fc0 sc0 ls0 wsd">Algebra v<span class="blank _0"></span>etorial gerada p<span class="blank _f"> </span>elo v<span class="blank _0"></span>etor <span class="ff7">u</span></div><div class="t m0 x29 he y76 ff6 fs0 fc0 sc0 ls0 ws51">N H<span class="ff2 wsa">[0<span class="ff7 ls4">,</span><span class="wsd">1]<span class="blank _2c"> </span>Conjun<span class="blank _0"></span>to de to<span class="blank _f"> </span>das as fun¸<span class="blank _d"></span>c˜<span class="blank _b"></span>oes de [0<span class="ff7 ls1">,</span>1] em <span class="ff9 ls16">R</span>que s˜<span class="blank _e"></span>ao H¨<span class="blank _b"></span>older em lugar algum</span></span></div><div class="t m0 x29 h14 y77 ff6 fs0 fc0 sc0 ls0 ws51">N H<span class="ffa fs5 ls1a v4">1</span><span class="ff2 wsa">[0<span class="ff7 ls1">,</span><span class="wsd">1]<span class="blank _2d"> </span>Conjun<span class="blank _0"></span>to de to<span class="blank _f"> </span>das as fun¸<span class="blank _d"></span>c˜<span class="blank _b"></span>oes de [0<span class="ff7 ls1">,</span>1] em <span class="ff9 ls16">R</span>que s˜<span class="blank _e"></span>ao Lipschitz em ponto algum</span></span></div><div class="t m0 x29 h15 y78 ff7 fs0 fc0 sc0 ls0 ws4a">m<span class="ffc fs5 ls1b v4">∗</span><span class="ff2 wsa">(</span>A<span class="ff2 wsd">)<span class="blank _2e"> </span>medida exterior do conjun<span class="blank _0"></span>to <span class="ff7">A</span></span></div><div class="t m0 x29 h2 y79 ff7 fs0 fc0 sc0 ls0 ws4a">m<span class="ff2 ls10">(</span>A<span class="ff2 wsd">)<span class="blank _2f"> </span>medida de Leb<span class="blank _f"> </span>esgue do conjun<span class="blank _0"></span>to <span class="ff7">A</span></span></div><div class="t m0 x11 h2 y8 ff2 fs0 fc0 sc0 ls0">1</div></div><div class="pi" data-data="{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}"></div></div>
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