Local Cᵏ Diffeomorphisms

In the setting of Functions of Class Cᵏ: Notation, let $k\in\mathbb{N}$ and let $f:U\to\mathbb{R}^{n}$.

(i) At a point. Let $a\in U$. $f$ is a local $C^{k}$ diffeomorphism at $a$ if there is an open set $W\subseteq\mathbb{R}^{n}$ with $a\in W\subseteq U$ such that $f(W)$ is open and the restriction $f|_{W}:W\to\mathbb{R}^{n}$ is a $C^{k}$ diffeomorphism of $W$ onto $f(W)$; it is a local smooth diffeomorphism at $a$ if the same holds with a smooth diffeomorphism.

(ii) On $U$. $f$ is a local $C^{k}$ diffeomorphism, respectively a local smooth diffeomorphism, if it is one at every point of $U$.