Elementarni aspekti neprekidnosti

master's thesis

• Title (english): Elementary aspects of continuity
• Author: Lea Lujo
• Mentor: Zvonko Iljazović (mentor)
• Committee member: Zvonko Iljazović (predsjednik povjerenstva)
• Committee member: Vedran Čačić (član povjerenstva)
• Committee member: Marko Horvat (član povjerenstva)
• Committee member: Marko Vrdoljak (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2022-10-07, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Ovaj diplomski rad podijelili smo na tri veća poglavlja u kojima smo promatrali neke elementarne aspekte neprekidnosti. U prvom poglavlju smo definirali neprekidne funkcije te proučavali nizove. Zatim smo promatrali omeđene podskupove od R te neka svojstva nizova, a uveli smo i pojam limesa funkcije te ispitivali njegovu vezu s neprekidnošću. Na kraju prvog poglavlja smo naveli aksiom potpunosti realnih brojeva. U drugom poglavlju smo se bavili neprekidnim funkcijama na segmentu. Dokazali smo da neprekidna funkcija na segmentu poprima sve međuvrijednosti te uveli pojmove gomilišta i skupilišta niza. Također smo dokazali da svaki omeđen niz realnih brojeva ima konvergentan podniz i da je svaka neprekidna funkcija na segmentu omeđena. U trećem poglavlju proučavali smo neprekidnost u kontekstu metričkih prostora. Na početku smo iskazali definiciju metričkog prostora i neke osnovne primjere, a zatim smo se bavili otvorenim skupovima u metričkom prostoru. Naposljetku smo proučavali neprekidne funkcije između metričkih prostora.

Abstract (english)

We divided this thesis into three larger chapters in which we observed some elementary aspects of continuity. In the first chapter, we defined continuous functions and studied sequences. Then we considered bounded subsets of R and some properties of sequences, and we also introduced the notion of the limit of a function and examined its connection with continuity. At the end of the first chapter, we stated the axiom of completeness of real numbers. In the second chapter we dealt with continuous functions on a segment. We proved that a continuous function on a segment takes on all intermediate values and introduced the notions of an accumulation point of a sequence. We also proved that every bounded sequence of real numbers has a convergent subsequence and that every continuous function on a segment is bounded. In the third chapter we studied continuity in the context of metric spaces. At the beginning, we stated the definition of the metric space and some basic examples, and then we dealt with open sets in the metric space. At the end, we observed continuous functions between metric spaces.

• Keywords: neprekidne funkcije; nizovi; limes funkcije; aksiom potpunosti realnih brojeva; gomilišt niza; skupilište niza; metrički prostori
• Keywords (english): continuous functions; sequences; limit of a function; axiom of completeness of real numbers; metric spaces
• Language: croatian
• URN:NBN: urn:nbn:hr:217:078113
• Study programme: Title: Mathematics Education; specializations in: Mathematics Education; Course: Mathematics Education; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra edukacije matematike (magistar/magistra edukacije matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2022-10-28 11:14:59