Local diffeomorphism corollary

Let $U\subseteq\mathbb{R}^n$ be open and let $f:U\to\mathbb{R}^n$ be $C^1$ . Suppose $a\in U$ and $\det Df(a)\neq 0$.

Corollary: There exist open neighborhoods $U_0$ of $a$ and $V_0$ of $f(a)$ such that $f:U_0\to V_0$ is a $C^1$ diffeomorphism (bijective , $C^1$, with $C^1$ inverse).

Connection to parent theorem: This is the inverse function theorem , often summarized as “$f$ is a local diffeomorphism at $a$ when $\det Df(a)\neq 0$.”