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[{"key": "dc.contributor.advisor", "value": "Kilpel\u00e4inen, Tero", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Lehtola, Piia", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2018-10-31T06:21:02Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2018-10-31T06:21:02Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2018", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/60048", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4ss\u00e4 tutkielmassa k\u00e4sitell\u00e4\u00e4n derivaattaa ja siihen liittyvi\u00e4 ilmi\u00f6it\u00e4. Aluksi k\u00e4yd\u00e4\u00e4n l\u00e4pi derivaatan ja jatkuvuuden yhteytt\u00e4, mit\u00e4 on tutkittu matematiikassa paljon. Jo 1800-luvulla osoitettiin, ett\u00e4 on olemassa jatkuva funktio, joka ei ole miss\u00e4\u00e4n pisteess\u00e4 derivoituva. Kuitenkin funktion derivoituvuudesta seuraa funktion jatkuvuus. T\u00e4t\u00e4 ei pid\u00e4 sekoittaa funktion derivaattafunktion jatkuvuuteen, sill\u00e4 derivaattafunktiot eiv\u00e4t v\u00e4ltt\u00e4m\u00e4tt\u00e4 ole jatkuvia. Niill\u00e4 on kuitenkin vastaava ominaisuus kuin jatkuvilla funktioilla, eli v\u00e4liss\u00e4olevien arvojen olemassolo. T\u00e4st\u00e4 seuraa, ett\u00e4 derivaattafunktiolla voi olla vain oleellisia ep\u00e4jatkuvuuspisteit\u00e4, eli pisteit\u00e4, joissa derivaattafunktion raja-arvoa ei ole olemassa tai se on \u00e4\u00e4ret\u00f6n.\n\nFunktiot eiv\u00e4t ole aina derivoituvia. T\u00e4st\u00e4 syyst\u00e4 on kehitetty yleistyksi\u00e4 perinteisest\u00e4 derivaatasta. T\u00e4ss\u00e4 tutkielmassa esitell\u00e4\u00e4n niist\u00e4 Dinin derivaatat ja funktion johdos. N\u00e4iden avulla pystyt\u00e4\u00e4n osoittamaan mahdollisesti derivoitumattomilla funktioilla vastaavanlaisia lauseita kuin perinteisell\u00e4 derivaatalla. \n\nTietyt ominaisuudet funktioilla takaavat kuitenkin derivoituvuuden melkein kaikkialla niiden m\u00e4\u00e4rittelyjoukossa. T\u00e4llaisia ominaisuuksia ovat monotonisuus, rajoitetusti heilahtelevuus ja absoluuttinen jatkuvuus. Funktion johdoksien avulla voidaan osoittaa, ett\u00e4 monotoniset funktiot ovat melkein kaikkialla derivoituvia. T\u00e4st\u00e4 ominaisuudesta seuraa, ett\u00e4 my\u00f6s rajoitetusti heilahtelevat funktiot ja absoluuttisesti jatkuvat funktiot ovat melkein kaikkialla derioituvia.\n\nTutkielman lopussa k\u00e4sitell\u00e4\u00e4n derivaatan integroimista ja osoitetaan, ett\u00e4 jos funktio halutaan saada takaisin sen derivaattafunktiota integroimalla, on funktion oltava t\u00e4ll\u00f6in absoluuttisesti jatkuva. Vastaesimerkkin\u00e4 toimii kuuluisa Cantorin funktio.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2018-10-31T06:21:02Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2018-10-31T06:21:02Z (GMT). No. of bitstreams: 0\n Previous issue date: 2018", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "51", "language": "", "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.format.mimetype", "value": "application/pdf", "language": null, "element": "format", "qualifier": "mimetype", "schema": "dc"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.subject.other", "value": "derivaatta", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "derivoituvuus", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Derivaatasta ja derivoituvuudesta", "language": "", "element": "title", "qualifier": null, "schema": "dc"}, {"key": "dc.type", "value": "master thesis", "language": null, "element": "type", "qualifier": null, "schema": "dc"}, {"key": "dc.identifier.urn", "value": "URN:NBN:fi:jyu-201810314561", "language": "", "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Faculty of Sciences", "language": "en", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Matematiikan ja tilastotieteen laitos", "language": "fi", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Department of Mathematics and Statistics", "language": "en", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikan opettajankoulutus", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Teacher education programme in Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "yvv.contractresearch.funding", "value": "0", "language": "", "element": "contractresearch", "qualifier": "funding", "schema": "yvv"}, {"key": "dc.type.coar", "value": "http://purl.org/coar/resource_type/c_bdcc", "language": null, "element": "type", "qualifier": "coar", "schema": "dc"}, {"key": "dc.rights.accesslevel", "value": "openAccess", "language": null, "element": "rights", "qualifier": "accesslevel", "schema": "dc"}, {"key": "dc.type.publication", "value": "masterThesis", "language": null, "element": "type", "qualifier": "publication", "schema": "dc"}, {"key": "dc.subject.oppiainekoodi", "value": "4041", "language": "", "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.subject.yso", "value": "derivoiminen", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "funktiot", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "differentiaalilaskenta", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.format.content", "value": "fulltext", "language": null, "element": "format", "qualifier": "content", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}, {"key": "dc.type.okm", "value": "G2", "language": null, "element": "type", "qualifier": "okm", "schema": "dc"}]
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