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)\neq 0\qquad\text{for every }x\in U,

then for each $x_0\in U$ there are neighborhoods $A$ of $x_0$ and $B$ of $f(x_0)$ such that $f|_A:A\to B$ is a diffeomorphism.

Remarks

This is the inverse function theorem applied at each point. In this situation, $f$ is called a local diffeomorphism.