summaryrefslogtreecommitdiff
path: root/macros/latex/contrib/stex/example/paper/continuous.tex
blob: dc90911f54b5f5e427bb1a6055f3ad36dfa750b9 (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
\begin{omgroup}[id=sec.contfuncs]{Continuous Functions}
\begin{module}[id=continuous]
\importmodule[load=\backmods{functions}]{functions}
\importmodule[load=\backmods{reals}]{reals}
\symdef{continuousfunctions}[2]{\mathcal{C}^0(#1,#2)}
\abbrdef{ContRR}[2]{\continuousfunctions\RealNumbers\RealNumbers}
\begin{definition}[for=continuousfunctions]
  A function $\fun{f}\RealNumbers\RealNumbers$ is called {\defi{continuous}} at
  $\inset{x}\RealNumbers$, iff for all $\epsilon>0$ there is a $\delta>0$, such that
 $\absval{f(x)-f(y)}<\epsilon$ for all $\absval{x-y}<\delta$. It is called
  {\defii{continuous}{on}} a set $\sseteq{S}\RealNumbers$, iff is is continous at all
  $\inset{x}S$, the set of all such functions is denoted with $\continuousfunctions{S}T$,
  if $\sseteq{f(S)}T$.
\end{definition}
\end{module}
\end{omgroup}
%%% Local Variables: 
%%% mode: LaTeX
%%% TeX-master: "paper"
%%% End: