Local $C^1$ Diffeomorphism on a Euclidean Open Set

Let $n$ be a natural number, let $U$ be an open subset of Euclidean space $\mathbb{R}^n$, and let $f:U\to\mathbb{R}^n$ be a $C^1$ map. Let $a\in U$. We say that $f$ is a local $C^1$ diffeomorphism at $a$ if there exist open sets $V,W\subseteq\mathbb{R}^n$ with $a\in V\subseteq U$ and $f(a)\in W$ such that $f(V)=W$, the restriction $f|_V:V\to W$ is bijective, and its inverse $(f|_V)^{-1}:W\to V$ is of class $C^1$.

We say $f$ is a local $C^1$ diffeomorphism (on $U$) if $f$ is a local $C^1$ diffeomorphism at every $a\in U$.