Let $U$ be an open subset of $\mathbb R^n$, let $F \in C^2(U;\mathbb R^n)$, and let $x_0 \in U$. If the Jacobian matrix $JF_{x_0}$ is invertible, then there are open neighbourhoods $U_0$ of $x_0$ contained in $U$ and $V_0$ of $F(x_0)$ contained in $\mathbb R^n$ such that $F: U_0 \to V_0$ is bijective and $F^{-1}: V_0 \to U_0$ belongs to $C^2(V_0;\mathbb R^n)$.