Množice nezveznosti realnih funkcij

diplomsko delo

Anja Kos (Author), Marko Slapar (Mentor)

Abstract

V diplomskem delu se ukvarjamo predvsem z vprašanjem, katere podmnožice realnih števil so lahko množice točk nezveznosti neke realne funkcije ene spremenljivke. Pokažemo, da je množica točk nezveznosti vedno števna unija zaprtih množic, kar na primer pomeni, da ne obstaja realna funkcija, ki bi bila nezvezna natanko na množici iracionalnih števil. To pokažemo s pomočjo Bairovega izreka o kategorijah. Na koncu diplomskega dela pokažemo, da je limita po točkah zaporedja zveznih funkcij vedno zvezna na precej veliki množici.

Keywords

zvezne funkcije;funkcije z omejeno variacijo;goste množice;nikjer goste množice;Bairov izrek;množice nezveznosti funkcij;

Data

Language:	Slovenian
Year of publishing:	2018
Typology:	2.11 - Undergraduate Thesis
Organization:	UL PEF - Faculty of Education
Publisher:	[A. Kos]
UDC:	51(043.2)
COBISS:	12149577
Views:	415
Downloads:	78
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Discontinuity sets of real functions
Secondary abstract:	In this diploma thesis, we are interested in understanding which subsets of real numbers can be sets of discontinuity of a real function of one variable. We show that any set of discontinuity is a countable union of closed sets, which, for example, excludes the possibility of an existence of a real function that is discontinuous precisely at irrational numbers. This is shown as an application of the Baire category theorem. In the last part of the thesis we show that the pointwise limit of a sequence of continuous functions is always continuous on a large subset of real numbers.
Secondary keywords:	mathematics;matematika;
File type:	application/pdf
Type (COBISS):	Bachelor thesis/paper
Thesis comment:	Univ. v Ljubljani, Pedagoška fak., Dvopredmetni učitelj: Matematika in računalništvo
Pages:	21 str.
ID:	10973399