Local diffeomorphism corollary

Local diffeomorphism corollary: Let $U\subseteq\mathbb R^k$ be an open set and let $f:U\to\mathbb R^k$ be continuously differentiable. If $\det Df(x)\ne 0$ for every $x\in U$, then for each $x_0\in U$ there exist neighborhoods $A$ of $x_0$ and $B$ of $f(x_0)$ such that the restriction $f:A\to B$ is a diffeomorphism .

This is an immediate consequence of the inverse function theorem applied at each point, and it is the standard meaning of saying that $f$ is a “local diffeomorphism.”