The Inverse Function Theorem

In the setting of Functions of Class Cᵏ: Notation, let $f:U\to\mathbb{R}^{n}$ be of class $C^{1}$ on $U$, and let $a\in U$ be a point at which the Jacobian matrix $Df(a)\in\mathcal{M}_{n}(\mathbb{R})$ is invertible.

(i) Local inverse. There is an open set $W\subseteq\mathbb{R}^{n}$ with $a\in W\subseteq U$ such that $V=f(W)$ is open, $f$ is injective on $W$, $Df(x)$ is invertible for every $x\in W$, and the inverse function $g:V\to W$ of $f|_{W}$, regarded as a function $V\to\mathbb{R}^{n}$, is of class $C^{1}$ on $V$, with

$$Dg\bigl(f(x)\bigr)=Df(x)^{-1}\qquad\text{for every }x\in W .$$

(ii) Higher regularity. Let $k\in\mathbb{N}$, and let $f$ be of class $C^{k}$ on $U$. Let $W\subseteq U$ be any open set with $f(W)$ open and $f$ injective on $W$ such that the inverse function $g:f(W)\to W$ of $f|_{W}$ is of class $C^{1}$ on $f(W)$, for instance a set $W$ as in (i). Then $g$ is of class $C^{k}$ on $f(W)$, and $f|_{W}$ is a $C^{k}$ diffeomorphism of $W$ onto $f(W)$. Moreover, as $Df(a)$ is invertible, $f$ is a local $C^{k}$ diffeomorphism at $a$.