Inverse function theorem

Let $\mathcal{O}\subset {\mathbb{R}}^n$ be open and $F:\mathcal{O}\to {\mathbb{R}}^n$ be continuously differentiable. Let $x_{\ast }\in \mathcal{O}$ be a point where $DF(x_{\ast })$ is invertible.

Then there is a neighborhood $U$ of $x_{\ast }$ and a neighborhood $V$ of $F(x_{\ast })$ such that the restriction $F{|}_U:U\to V$

is a bijection and $F^{-1}:V\to U$ is continuously differentiable and for $y\in V$ with $F(x)=y$ then

$D{F}^{-1}(y)=[DF(x){]}^{-1}$