I am trying to work out if the following variant of the inverse function theorem holds:
Conjecture: Suppose $f$ is a $C^0$ function, and $f'(0)$ exists and is non-singular. Then there exists a neighbourhood around $0$ over which $f$ is a homeomorphism (i.e. continuous with a continuous inverse).
This differs from the commonly used variants of the theorem as $f'$ is not assumed to exist in a neighbourhood of $0$, only at $0$ itself. I believe this holds but cannot find a good reference for it. Please could someone provide either a reference or a counterexample?