2. Let $\mathcal{O}$ be an open set of $\mathbb{R}^n$ and let $f : \mathcal{O} \rightarrow \mathbb{R}^n$ be a map of class $C^1$ whose differential at $x$ is invertible for all $x \in \mathcal{O}$. Prove that the image by $f$ of an open set of $\mathcal{O}$ is an open set of $\mathbb{R}^n$.