Definition:Local Diffeomorphism

Definition
Let $n$ and $k$ be natural numbers.

Let $U \subset \R^n$ be an open set.

Let $f: U \to \R^n$ be a mapping.

Then $f$ is a local $C^k$-diffeomorphism every $a \in U$ has a open neighborhhood such that the restriction of $f$ to it is a $C^k$-diffeomorphism on its image.