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\QTR{small}{Matematick\U{e1} anal\U{fd}za II online - Integr\U{e1}lny po\U{10d}et - Nutn\U{e1} \ posta\U{10d}uj\U{fa}ca podmienka integrovate\U{13e}nosti\dotfill \thepage }}
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\input{tcilatex}
\begin{document}
\author{A. U. Thor}
\title{Lab Report}
\date{The Date }
\maketitle
\begin{abstract}
A Laboratory report created with Scientific Notebook
\end{abstract}
\section{Integr\'{a}lny po\v{c}et}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
\textbf{%
%TCIMACRO{\hyperref{Obsah}{}{}{maiindex.tex}}%
%BeginExpansion
\msihyperref{Obsah}{}{}{maiindex.tex}%
%EndExpansion
} & \textbf{%
%TCIMACRO{\hyperref{Obsah kapitoly}{}{}{Ma5.tex}}%
%BeginExpansion
\msihyperref{Obsah kapitoly}{}{}{Ma5.tex}%
%EndExpansion
} & \textbf{%
%TCIMACRO{\hyperref{Predch\'{a}dzaj\'{u}ca str\'{a}nka}{}{}{Ma51.tex}}%
%BeginExpansion
\msihyperref{Predch\'{a}dzaj\'{u}ca str\'{a}nka}{}{}{Ma51.tex}%
%EndExpansion
} & \textbf{%
%TCIMACRO{\hyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{Ma53.tex}}%
%BeginExpansion
\msihyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{Ma53.tex}%
%EndExpansion
} & \textbf{%
%TCIMACRO{\hyperref{Ot\'{a}zky}{}{}{O5.tex}}%
%BeginExpansion
\msihyperref{Ot\'{a}zky}{}{}{O5.tex}%
%EndExpansion
} & \textbf{%
%TCIMACRO{\hyperref{Cvi\v{c}enia}{}{}{C5.tex}}%
%BeginExpansion
\msihyperref{Cvi\v{c}enia}{}{}{C5.tex}%
%EndExpansion
} & \textbf{%
%TCIMACRO{\hyperref{Index}{}{}{Glos.tex}}%
%BeginExpansion
\msihyperref{Index}{}{}{Glos.tex}%
%EndExpansion
} \\ \hline
\end{tabular}
\end{center}
\subsection{Nutn\'{a} a posta\v{c}uj\'{u}ca podmienka integrovate\v{l}nosti.}
Teraz budeme sk\'{u}ma\v{t} mno\v{z}iny bodov nespojitosti funkcie.
\begin{definition}
\emph{Oscil\'{a}cia} ohrani\v{c}enej funkcie $f:S\left( \subset \mathbf{R}%
^{n}\right) \longrightarrow \mathbf{R}$ na $S$ je definovan\'{a}
\[
oscf(S)=\sup_{\mathbf{x},\mathbf{y}\in S}\left\{ f(\mathbf{x})-f(\mathbf{y}%
)\right\} .
\]
\end{definition}
Pre oscil\'{a}ciu plat\'{\i}
\[
oscf(S)=\sup_{\mathbf{x}\in S}\left\{ f(x)\right\} -\inf_{\mathbf{y}\in
S}\left\{ f(\mathbf{y})\right\} ,
\]
\v{l}ahko si mo\v{z}no overi\v{t}, \v{z}e
\[
oscf(S_{1})\leq oscf(S_{2})\text{ \ ak \ \ }S_{1}\subseteq S_{2}.
\]
\begin{definition}
\emph{Oscil\'{a}ciou }$f$\emph{\ v bode} $\mathbf{x\in }S$ \ budeme rozumie%
\v{t} \v{c}\'{\i}slo
\[
\omega _{f(\mathbf{x})}=\inf \left\{ oscf(O_{\delta }(\mathbf{x})\cap
S);\delta >0\right\} =\lim_{\delta \longrightarrow 0^{+}}oscf\left(
O_{\delta }(x)\cap S\right) .
\]
\end{definition}
Funkcia $oscf(O_{\delta }(x)\cap S)$ je klesaj\'{u}ca funkcia premennej $%
\delta .$
\begin{theorem}
Ohrani\v{c}en\'{a} funkcia $f:S\left( \subset \mathbf{R}^{n}\right)
\longrightarrow \mathbf{R}$ je spojit\'{a} v bode $\mathbf{a}\in S$ vtedy a
len vtedy ak $\omega _{f(\mathbf{a})}=0.$
\end{theorem}
\begin{tabular}{|c|}
\hline
{\small
%TCIMACRO{\hyperref{D\^{o}kaz}{}{}{DO521.tex}}%
%BeginExpansion
\msihyperref{D\^{o}kaz}{}{}{DO521.tex}%
%EndExpansion
} \\ \hline
\end{tabular}%
\label{4}
Ozna\v{c}me $D=\left\{ \mathbf{x\in }E;\omega _{f(\mathbf{x})}>0\right\} $
mno\v{z}inu bodov nespojitosti funkcie $f$ na n-kv\'{a}dri $E$. Vieme, \v{z}%
e $f$ je integrovate\v{l}n\'{a}, ak existuje delenie $\mathcal{P}$ n-kv\'{a}%
dra $E$\ tak\'{e}, \v{z}e rozdiel%
\[
H\left( f,\mathcal{P}\right) -D\left( f,\mathcal{P}\right)
=\sum_{i=1}^{p}(M_{i}-m_{i})c(E_{i})
\]%
sa d\'{a} urobi\v{t} dostato\v{c}ne mal\'{y}. Na $n$-kv\'{a}droch $E_{i}$,
kde je funkcia $f$ spojit\'{a} mo\v{z}no v\'{y}raz $M_{i}-m_{i}=oscf(E_{i})$
urobi\v{t} dostato\v{c}ne mal\'{y}. A preto\v{z}e predpoklad\'{a}me, \v{z}e
funkcia $f$ je ohrani\v{c}en\'{a}, tak aj cel\'{u} sumu mo\v{z}no urobi\v{t}
malou.
\begin{definition}
\label{9}Nech $A\subset \mathbf{R}^{n}$ je ohrani\v{c}en\'{a} mno\v{z}ina.
Hovor\'{\i}me, \v{z}e $A$ m\'{a} $n$\emph{-dimenzion\'{a}lny Jordanovsk\'{y}
obsah nula,} ak $\forall \varepsilon >0$ existuje kone\v{c}n\'{e} pokrytie
mno\v{z}iny $A$ pomocou $n$-kv\'{a}drov $\left\{ E_{k};\,k=1,2,\dots
,q\right\} $ (t.j. $A\subset \bigcup_{i=1}^{q}E_{i}$) tak, \v{z}e
\[
\sum_{i=1}^{q}c(E_{i})<\varepsilon .
\]%
Hovor\'{\i}me, \v{z}e $A$ m\'{a} $n$\emph{-dimenzion\'{a}lnu Lebesquovu
mieru nula,} ak $\forall \varepsilon >0$ existuje spo\v{c}\'{\i}tate\v{l}n%
\'{e} pokrytie mno\v{z}iny $A$ pomocou syst\'{e}mu $n$-kv\'{a}drov $\left\{
E_{i};\,i=1,2,\dots ,\right\} $(t.j. $A\subset \bigcup_{i=1}^{\infty }E_{i}$%
) tak, \v{z}e
\[
\sum_{i=1}^{\infty }c(E_{i})<\varepsilon .
\]
\end{definition}
\begin{description}
\item[Pozn\'{a}mka] \label{2}Je jasn\'{e}, \v{z}e ak $A\subset \mathbf{R}%
^{n} $ m\'{a} Jordanovsk\'{y} obsah nula, tak m\'{a} aj Lebesquovu mieru
nula, ale \textsl{opa\v{c}n\'{e} tvrdenie neplat\'{\i}.} Pre \emph{kompaktn%
\'{e} mno\v{z}iny je} mno\v{z}ina s obsahom nula a mno\v{z}ina s mierou nula
to ist\'{e}.
\end{description}
Poznamenajme, \v{z}e \'{u}se\v{c}ka v $\mathbf{R}^{2}$ m\'{a} dvojdimenzion%
\'{a}lnu mieru $0,$ aj ke\v{d} jej jednodimenzion\'{a}lna miera (d\'{l}\v{z}%
ka) je nenulov\'{a}. Je to preto, \v{z}e \'{u}se\v{c}ku m\^{o}\v{z}eme pokry%
\v{t} obd\'{l}\v{z}nikmi s kone\v{c}nou d\'{l}\v{z}kou a s ve\v{l}mi malou
\v{s}\'{\i}rkou.
\begin{example}
Nech $f:\left\langle a,b\right\rangle \longrightarrow \mathbf{R}$ je spojit%
\'{a}. Potom jej graf $G=\left\{ \left( x,f\left( x\right) )\right) \mathbf{%
\in R}^{2};x\mathbf{\in }\left\langle a,b\right\rangle \right\} $ m\'{a}
dvojdimenzion\'{a}lny obsah 0.
\end{example}
\begin{solution}
Nech $\varepsilon >0$ je \v{l}ubovo\v{l}n\'{e}. Preto\v{z}e $f$ je aj
rovnomerne spojit\'{a} na $\left\langle a,b\right\rangle $.
\[
\forall \varepsilon >0\,\exists \delta >0;\,\left| x-y\right| <\delta
\Longrightarrow \left| f(x)-f(y)\right| <\varepsilon .
\]%
Nech $a=x_{0}