Continuously differentiable maps, local inverses, and local diffeomorphisms

Let $U\subseteq\mathbb R^m$ be open and $f:U\to\mathbb R^n$. The map $f$ is continuously differentiable, or of class $C^1$, when it is totally differentiable at every point of $U$ and the entries of its derivative matrix are continuous functions on $U$.

For an open $U\subseteq\mathbb R^n$ and $a\in U$, a local inverse of $f$ at $a$ is a function $g:W\to V$ for open neighbourhoods $a\in V\subseteq U$ and $f(a)\in W\subseteq\mathbb R^n$ such that $f|_V:V\to W$ is bijective and $g=(f|_V)^{-1}$. If both $f|_V$ and $g$ are $C^1$, this restriction is a local diffeomorphism at $a$.