\begin{omgroup}[id=sec.contfuncs]{Continuous Functions}
\begin{module}[id=continuous]
\importmodule[\backmods{functions}]{functions}
\importmodule[\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}
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