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+#LyX 2.4 created this file. For more info see https://www.lyx.org/
+\lyxformat 620
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass article
+\begin_preamble
+\usepackage{hyperref}
+\usepackage{siunitx}
+\usepackage{pgfplots}
+\usepackage{listings}
+\usepackage{multicol}
+\sisetup{output-decimal-marker = {,}, quotient-mode=fraction, output-exponent-marker=\ensuremath{\mathrm{3}}}
+\usepackage{amsmath}
+\usepackage{tikz}
+\newcommand{\udensdash}[1]{%
+    \tikz[baseline=(todotted.base)]{
+        \node[inner sep=1pt,outer sep=0pt] (todotted) {#1};
+        \draw[densely dashed] (todotted.south west) -- (todotted.south east);
+    }%
+}%
+\DeclareMathOperator{\Lin}{Lin}
+\DeclareMathOperator{\rang}{rang}
+\DeclareMathOperator{\sled}{sled}
+\DeclareMathOperator{\Aut}{Aut}
+\DeclareMathOperator{\red}{red}
+\DeclareMathOperator{\karakteristika}{char}
+\usepackage{algorithm,algpseudocode}
+\providecommand{\corollaryname}{Posledica}
+\end_preamble
+\use_default_options true
+\begin_modules
+enumitem
+theorems-ams
+\end_modules
+\maintain_unincluded_children no
+\language slovene
+\language_package default
+\inputencoding auto-legacy
+\fontencoding auto
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_roman_osf false
+\font_sans_osf false
+\font_typewriter_osf false
+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry true
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification false
+\use_refstyle 1
+\use_formatted_ref 0
+\use_minted 0
+\use_lineno 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\leftmargin 2cm
+\topmargin 2cm
+\rightmargin 2cm
+\bottommargin 2cm
+\headheight 2cm
+\headsep 2cm
+\footskip 1cm
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style german
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tablestyle default
+\tracking_changes false
+\output_changes false
+\change_bars false
+\postpone_fragile_content false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\docbook_table_output 0
+\docbook_mathml_prefix 1
+\end_header
+
+\begin_body
+
+\begin_layout Title
+ANA2 IŠRM 2023/24
+\end_layout
+
+\begin_layout Author
+
+\noun on
+Anton Luka Šijanec
+\end_layout
+
+\begin_layout Date
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+today
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Množice v 
+\begin_inset Formula $\mathbb{R}^{n}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Definition*
+Razdalja točk v 
+\begin_inset Formula $\mathbb{R}^{n}$
+\end_inset
+
+ je norma njune razlike.
+ 
+\begin_inset Formula $\varepsilon-$
+\end_inset
+
+okolica točke 
+\begin_inset Formula $a\in\mathbb{R}^{n}$
+\end_inset
+
+ so take točke,
+ ki so od 
+\begin_inset Formula $a$
+\end_inset
+
+ oddaljene manj od 
+\begin_inset Formula $\varepsilon\in\mathbb{R}$
+\end_inset
+
+.
+ Robna točka množice 
+\begin_inset Formula $A$
+\end_inset
+
+ je taka točka,
+ katere poljubno majhna okolica vsebuje tako točke iz 
+\begin_inset Formula $A$
+\end_inset
+
+ kot tudi točke,
+ ki niso iz 
+\begin_inset Formula $A$
+\end_inset
+
+.
+ Odprta množica ne vsebuje robnih točk.
+ Zaprta množica je komplement neke odprte množice.
+\end_layout
+
+\begin_layout Claim*
+\begin_inset Formula $A\subset\mathbb{R}$
+\end_inset
+
+ zaprta 
+\begin_inset Formula $\Leftrightarrow$
+\end_inset
+
+ za vsako zaporedje s členi v 
+\begin_inset Formula $A$
+\end_inset
+
+ velja,
+ da so vsa njegova stekališča,
+ čim obstajajo,
+ v 
+\begin_inset Formula $A$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Proof
+Dokazujemo ekvivalenco
+\end_layout
+
+\begin_deeper
+\begin_layout Description
+\begin_inset Formula $\left(\Rightarrow\right)$
+\end_inset
+
+ Naj bo 
+\begin_inset Formula $s$
+\end_inset
+
+ stekališče 
+\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$
+\end_inset
+
+,
+ 
+\begin_inset Formula $a_{n}\in A$
+\end_inset
+
+ in 
+\begin_inset Formula $s\not\in A$
+\end_inset
+
+ (RAAPDD).
+ Ker je 
+\begin_inset Formula $A$
+\end_inset
+
+ zaprta,
+ je 
+\begin_inset Formula $\mathbb{R}\setminus A$
+\end_inset
+
+ odprta,
+ zato 
+\begin_inset Formula $\exists\varepsilon>0\ni:\left(s-\varepsilon,s+\varepsilon\right)\subset\mathbb{R}\setminus A$
+\end_inset
+
+,
+ torej v 
+\begin_inset Formula $\left(s-\varepsilon,s+\varepsilon\right)$
+\end_inset
+
+ ni nobenega člena zaporedja,
+ torej 
+\begin_inset Formula $s$
+\end_inset
+
+ ni stekališče 
+\begin_inset Formula $\rightarrow\!\leftarrow$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Description
+\begin_inset Formula $\left(\Leftarrow\right)$
+\end_inset
+
+ Dokazujemo,
+ da je 
+\begin_inset Formula $A$
+\end_inset
+
+ zaprta,
+ torej,
+ da je 
+\begin_inset Formula $B=\mathbb{R}\setminus A$
+\end_inset
+
+ odprta.
+ PDDRAA 
+\begin_inset Formula $B$
+\end_inset
+
+ ni odprta 
+\begin_inset Formula $\Rightarrow\exists x\in B\ni:\forall n\in\mathbb{N}:$
+\end_inset
+
+ 
+\begin_inset Formula $n^{-1}-$
+\end_inset
+
+okolica 
+\begin_inset Formula $x$
+\end_inset
+
+ vsebuje nek element 
+\begin_inset Formula $A$
+\end_inset
+
+.
+ Našli smo torej zaporedje v 
+\begin_inset Formula $A$
+\end_inset
+
+ s stekališčem v 
+\begin_inset Formula $B$
+\end_inset
+
+.
+ 
+\begin_inset Formula $\rightarrow\!\leftarrow$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Definition*
+Stroga podmnožica 
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+ je kompaktna,
+ če ima vsako zaporedje s členi v njej v njej tudi stekališče.
+ Množica je omejena,
+ če je podmnožica neke okolice izhodišča.
+\end_layout
+
+\begin_layout Theorem*
+\begin_inset Formula $A\subset\mathbb{R}$
+\end_inset
+
+ kompaktna 
+\begin_inset Formula $\Leftrightarrow A$
+\end_inset
+
+ zaprta in omejena.
+\end_layout
+
+\begin_layout Proof
+Dokazujemo ekvivalenco
+\end_layout
+
+\begin_deeper
+\begin_layout Labeling
+\labelwidthstring 00.00.0000
+\begin_inset Formula $\left(\Leftarrow\right)$
+\end_inset
+
+ Naj bo 
+\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}}$
+\end_inset
+
+ zaporedje v 
+\begin_inset Formula $A$
+\end_inset
+
+.
+ Ker je 
+\begin_inset Formula $A$
+\end_inset
+
+ omejena,
+ je zaporedje omejeno,
+ torej premore stekališča.
+ Ker je 
+\begin_inset Formula $A$
+\end_inset
+
+ zaprta,
+ vsebuje vsa ta stekališča.
+ Torej je 
+\begin_inset Formula $A$
+\end_inset
+
+ kompaktna.
+\end_layout
+
+\begin_layout Labeling
+\labelwidthstring 00.00.0000
+\begin_inset Formula $\left(\Rightarrow\right)$
+\end_inset
+
+ 
+\begin_inset Formula $A$
+\end_inset
+
+ je omejeno,
+ sicer bi našli zaporedje,
+ da velja 
+\begin_inset Formula $a_{i}\geq i$
+\end_inset
+
+,
+ ki nima stekališča.
+ Treba je dokazati še,
+ da je 
+\begin_inset Formula $A$
+\end_inset
+
+ zaprta.
+ Vsa stekališča zaporedij s členi v 
+\begin_inset Formula $A$
+\end_inset
+
+ imajo v 
+\begin_inset Formula $A$
+\end_inset
+
+ stekališče (kompaktnost).
+ Torej za vsako stekališče zaporedja s členi v 
+\begin_inset Formula $A$
+\end_inset
+
+ velja,
+ da ima v 
+\begin_inset Formula $A$
+\end_inset
+
+ stekališče,
+ torej je 
+\begin_inset Formula $A$
+\end_inset
+
+ zaprta.
+\end_layout
+
+\end_deeper
+\begin_layout Remark*
+Vsako zaporedje v kompaktni množici ima stekališče,
+ kar za zaprto množico ni rečeno.
+ Zaprta množica lahko vsebuje zaporedja brez stekališč.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+razloži normo,
+ trikotniško neenakost,
+ itd.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Definition*
+Točka 
+\begin_inset Formula $a\in A\subseteq\mathbb{R}^{n}$
+\end_inset
+
+ je notranja,
+ če obstaja neka njena okolica,
+ ki je podmnožica 
+\begin_inset Formula $A$
+\end_inset
+
+.
+ Točka 
+\begin_inset Formula $a\in\mathbb{R}^{n}$
+\end_inset
+
+ je stekališče množice 
+\begin_inset Formula $A$
+\end_inset
+
+,
+ če vsaka njena okolica seka
+\begin_inset Foot
+status open
+
+\begin_layout Plain Layout
+t.
+ j.
+ ima neprazen presek z
+\end_layout
+
+\end_inset
+
+ 
+\begin_inset Formula $A\setminus\left\{ a\right\} $
+\end_inset
+
+.
+ Točka 
+\begin_inset Formula $a\in A$
+\end_inset
+
+,
+ ki ni stekališče 
+\begin_inset Formula $A$
+\end_inset
+
+,
+ je izolirana točka 
+\begin_inset Formula $A$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
+
+\end_layout
+
+\begin_layout Definition*
+Zaporedje s členi v 
+\begin_inset Formula $\mathbb{R}^{k}$
+\end_inset
+
+ je funkcija 
+\begin_inset Formula $\mathbb{N}\to\mathbb{R}^{k}$
+\end_inset
+
+,
+ 
+\begin_inset Formula $n\mapsto a_{n}=\left(a_{n}^{\left(1\right)},\dots,a_{n}^{\left(k\right)}\right)$
+\end_inset
+
+.
+ 
+\begin_inset Formula $a\in\mathbb{R}^{k}$
+\end_inset
+
+ je limita zaporedja 
+\begin_inset Formula $\left(a_{n}\right)_{n}$
+\end_inset
+
+ s členi v 
+\begin_inset Formula $\mathbb{R}^{k}$
+\end_inset
+
+,
+ če 
+\begin_inset Formula $\forall\varepsilon>0\exists n_{0}\in\mathbb{N}\forall n>n_{0}:\left|a-a_{n}\right|<\varepsilon$
+\end_inset
+
+ in pišemo 
+\begin_inset Formula $a=\lim_{n\to\infty}a_{n}$
+\end_inset
+
+.
+ Če zaporedje ima limito,
+ je konvergentno,
+ sicer je divergentno.
+ Točka 
+\begin_inset Formula $s\in\mathbb{R}^{k}$
+\end_inset
+
+ je stekališče zaporedja 
+\begin_inset Formula $\left(a_{n}\right)_{n}$
+\end_inset
+
+ s členi v 
+\begin_inset Formula $\mathbb{R}^{k}$
+\end_inset
+
+,
+ če je v vsaki okolici 
+\begin_inset Formula $s$
+\end_inset
+
+ neskončno členov 
+\begin_inset Formula $\left(a_{n}\right)_{n}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Fact*
+Velja:
+\end_layout
+
+\begin_deeper
+\begin_layout Itemize
+Vsako konvergentno zaporedje je omejeno in ima natanko eno limito,
+ ki je njegovo edino stekališče.
+\end_layout
+
+\begin_layout Itemize
+Vsako omejeno zaporedje ima stekališče.
+\end_layout
+
+\begin_layout Itemize
+Stekališče zaporedja je limita nekega podzaporedja in obratno.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $A\subset\mathbb{R}^{k}$
+\end_inset
+
+ je zaprta 
+\begin_inset Formula $\Leftrightarrow$
+\end_inset
+
+ vsako stekališče zaporedja s členi v 
+\begin_inset Formula $A$
+\end_inset
+
+ leži v 
+\begin_inset Formula $A$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Section
+Funkcije več spremenljivk
+\end_layout
+
+\begin_layout Definition*
+Naj bo 
+\begin_inset Formula $D\subseteq\mathbb{R}^{k}$
+\end_inset
+
+ in 
+\begin_inset Formula $f:D\to\mathbb{R}$
+\end_inset
+
+ preslikava.
+ Če je 
+\begin_inset Formula $k\geq2$
+\end_inset
+
+,
+ je 
+\begin_inset Formula $f$
+\end_inset
+
+ funkcija več spremenljivk.
+ 
+\begin_inset Formula $\Gamma_{f}=\left\{ \left(x,fx\right);x\in D\right\} \subset\mathbb{R}^{k}\times\mathbb{R}$
+\end_inset
+
+ je graf funkcije 
+\begin_inset Formula $f$
+\end_inset
+
+.
+ Za 
+\begin_inset Formula $a\in\mathbb{R}^{k}$
+\end_inset
+
+ stekališče 
+\begin_inset Formula $D$
+\end_inset
+
+ je 
+\begin_inset Formula $L\in\mathbb{R}$
+\end_inset
+
+ limita 
+\begin_inset Formula $f$
+\end_inset
+
+ v 
+\begin_inset Formula $a$
+\end_inset
+
+,
+ če 
+\begin_inset Formula $\forall\varepsilon>0\exists\delta=\delta\left(a,\varepsilon\right)>0\forall x\in D,x\not=a:\left|\left|x-a\right|\right|<\delta\Rightarrow\left|fx-L\right|<\varepsilon$
+\end_inset
+
+ in pišemo 
+\begin_inset Formula $\lim_{x\to a}fx=L$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Remark*
+Medtem ko imamo pri funkcijah ene spremenljivke levo in desno limito,
+ je tu obnašanje bolj zapleteno,
+ saj obstaja veliko različnih načinov približevanja k 
+\begin_inset Formula $a$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Definition*
+Naj bo 
+\begin_inset Formula $D\subseteq\mathbb{R}^{k}$
+\end_inset
+
+ in 
+\begin_inset Formula $f:D\to\mathbb{R}$
+\end_inset
+
+ funkcija in 
+\begin_inset Formula $a\in D$
+\end_inset
+
+.
+ 
+\begin_inset Formula $f$
+\end_inset
+
+ je zvezna v 
+\begin_inset Formula $a$
+\end_inset
+
+,
+ če 
+\begin_inset Formula $\forall\varepsilon>0\exists\delta=\delta\left(a,\varepsilon\right)>0\forall x\in D:\left|\left|x-a\right|\right|<\delta\Rightarrow\left|fx-fa\right|<\varepsilon$
+\end_inset
+
+.
+ 
+\begin_inset Formula $f$
+\end_inset
+
+ je zvezna,
+ če je zvezna na vsaki točki svojega definicijskega območja.
+\end_layout
+
+\begin_layout Remark*
+Če je 
+\begin_inset Formula $a$
+\end_inset
+
+ stekališče 
+\begin_inset Formula $D$
+\end_inset
+
+,
+ je 
+\begin_inset Formula $f$
+\end_inset
+
+ zvezna v 
+\begin_inset Formula $a\Leftrightarrow\lim_{x\to a}fx=fa$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Corollary*
+Če je 
+\begin_inset Formula $a$
+\end_inset
+
+ izolirana točka 
+\begin_inset Formula $D$
+\end_inset
+
+,
+ je 
+\begin_inset Formula $f$
+\end_inset
+
+ zvezna v 
+\begin_inset Formula $a$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Definition*
+Naj bo 
+\begin_inset Formula $f:D\subseteq\mathbb{R}^{k}\to\mathbb{R}$
+\end_inset
+
+ funkcija,
+ 
+\begin_inset Formula $Z=fD$
+\end_inset
+
+ njena zaloga vrednosti in 
+\begin_inset Formula $g:Z\to\mathbb{R}$
+\end_inset
+
+ funkcija.
+ Kompozitum ali sestavljena funkcija 
+\begin_inset Formula $f$
+\end_inset
+
+ in 
+\begin_inset Formula $g$
+\end_inset
+
+ je funkcija 
+\begin_inset Formula $k$
+\end_inset
+
+ spremenljivk 
+\begin_inset Formula $g\circ f:D\to\mathbb{R}$
+\end_inset
+
+,
+ definirana s predpisom 
+\begin_inset Formula $\left(g\circ f\right)x=gfx$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Theorem*
+Naj bo 
+\begin_inset Formula $f$
+\end_inset
+
+ funkcija 
+\begin_inset Formula $k$
+\end_inset
+
+ spremenljivk,
+ zvezna v 
+\begin_inset Formula $a\in\mathbb{R}^{k}$
+\end_inset
+
+ in 
+\begin_inset Formula $g$
+\end_inset
+
+ funkcija ene spremenljivke,
+ zvezna v 
+\begin_inset Formula $fa\in\mathbb{R}$
+\end_inset
+
+.
+ Tedaj je 
+\begin_inset Formula $g\circ f$
+\end_inset
+
+ zvezna v 
+\begin_inset Formula $a$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Proof
+Izberimo poljuben 
+\begin_inset Formula $\varepsilon$
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document