# An Inverse Function Theorem Converse

## TL;DR

This paper proves a converse to the inverse function theorem in Banach spaces, showing that under certain conditions, the inverse function inherits differentiability from the original function.

## Contribution

It establishes that if an inverse homeomorphism is locally Lipschitz and the original function is $C^p$, then the inverse is also $C^p$, providing a new perspective on inverse function regularity.

## Key findings

- The Fréchet derivative of $g$ is invertible at each point.
- The inverse function $f$ is differentiable of class $C^p$.
- The result extends inverse function theorem to less regular inverse functions.

## Abstract

We establish the following converse of the well-known inverse function theorem. Let $g:U\to V$ and $f:V\to U$ be inverse homeomorphisms between open subsets of Banach spaces. If $g$ is differentiable of class $C^p$ and $f$ if locally Lipschitz, then the Fr\'echet derivative of $g$ at each point of $U$ is invertible and $f$ must be differentiable of class $C^p$.