\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: