I'm having difficulty with the following question which was given to me following studying the inverse mapping theorem.
Let $U\subseteq\mathbb{R}^{n}$ be an open set and let $f:U\to\mathbb{R}^{n}$ be injective and differentiable in $U$ , assume also $f\left(U\right)$ is an open set and let $g:f\left(U\right)\to U$ be the inverse of $f$ . Prove that if $g$ is Lipschitzian then it is differentiable.
I'm assuming the main thing I'm missing is how to use the condition that $f(U)$ is open.
Help would be most appreciated!