Converse to Sard's theorem Combining the inverse function theorem and sard's theorem we arrive at: > $f: \mathbb{R}^n \to \mathbb{R}^n$ is smooth and locally injective $\implies$ f is a local diffeomorphism almost ev...--prophetes.ai

Converse to Sard's theorem Combining the inverse function theorem and sard's theorem we arrive at: > $f: \mathbb{R}^n \to \mathbb{R}^n$ is smooth and locally injective $\implies$ f is a local diffeomorphism almost everywhere Does a converse to this exist? In other words: Let $f: \mathbb{R}^n \to \mathbb{R}^n$ be a continuous locally injective map which is a local diffeomorphism almost everywhere. Must $f$ be smooth?

No, take for example $f(x)=x|x|$. Then $f$ is locally a diffeomorphism everywhere except at $x=0$, but not smooth.