An inverse function theorem converse

Abstract

We establish the following converse of the well-known inverse function theorem. Let g:U→V and f:V→U be inverse homeomorphisms between open subsets of Banach spaces. If g is differentiable of class Cp and f is locally Lipschitz, then the Fréchet derivative of g at each point of U is invertible and f must be differentiable of class Cp.