If $f:{\Bbb R}^m \to {\Bbb R}^n$ is Lipschitz on all compact sets, under what conditions is $f$ a $C^1$ map?
Background: I have proved the following result, and I am looking for a converse.
Proposition. Let $f: {\Bbb R}^m \to {\Bbb R}^n$ be a $C^1$ mapping and let $K \subset {\Bbb R}^m$ be compact. Then the restriction $f|_K$ of $f$ to $K$ is Lipschitz continuous.
The proof of this result, and some special cases, can be found here: Post 1, Post 2, Post 3, and Post 4.
In general, differentiability is a stronger condition than continuity, so there is no reason to expect the "Lipschitz on all compact sets" assumption above to imply differentiability. What other additional assumptions on $f$ would be required to ensure that it is a $C^1$ mapping?
Rademacher's Theorem is a related result; but I am clearly looking for something stronger.
The goal is to characterize $C^1$ maps in terms of Lipschitz continuity on compact sets, and other additional conditions if required.
Thank you, and I am excited to see where this goes!