\begin{definition}[Definition:Local Diffeomorphism]
Let $n$ and $k$ be natural numbers.
Let $U \subset \R^n$ be an open set.
{{explain|Context of Open Set}}
Let $f: U \to \R^n$ be a mapping.
Then $f$ is a '''local $C^k$-diffeomorphism''' {{iff}} every $a \in U$ has a open neighborhhood such that the restriction of $f$ to it is a $C^k$-diffeomorphism on its image.
\end{definition}