Cᵏ Diffeomorphisms Between Open Subsets of ℝⁿ

In the setting of Functions of Class Cᵏ: Notation, let $W$ and $V$ be open subsets of $\mathbb{R}^{n}$.

(i) Diffeomorphisms. Let $k\in\mathbb{N}$. A function $f:W\to\mathbb{R}^{n}$ is a $C^{k}$ diffeomorphism of $W$ onto $V$ if $f(W)=V$, $f$ is injective, $f$ is of class $C^{k}$ on $W$, and the inverse function $f^{-1}:V\to W$, regarded as a function $V\to\mathbb{R}^{n}$, is of class $C^{k}$ on $V$. Likewise, $f$ is a smooth diffeomorphism of $W$ onto $V$ if $f(W)=V$, $f$ is injective, $f$ is smooth on $W$, and $f^{-1}:V\to\mathbb{R}^{n}$ is smooth on $V$.