Past exam of the options I course of the University of Oxford / 2013 / F / 1 / c / ii / Solution

The inverse function theorem states: let $F:U\subseteq {\mathbb{R}}^n\to {\mathbb{R}}^n$ be $C^1$, with $U$ open, and suppose $DF(x_0)$ is invertible. Then there exist open neighbourhoods $U_0\subseteq U$ of $x_0$ and $V_0$ of $F(x_0)$ such that $F:U_0\to V_0$ is a $C^1$ diffeomorphism. Its inverse derivative satisfiesThe conclusion is local; invertibility of one Jacobian matrix does not assert global injectivity.