Hölder continuity and differentiability on subsequences

Abstract

It is shown that an arbitrary function from $D\subset {\mathbb{R}}^n$ to ${\mathbb{R}}^m$ will become $C^{0,\alpha }$-continuous in almost every $x\in D$ after restriction to a certain subset with limit point $x$. For $n\ge m$ differentiability can be obtained. Examples show the Hölder exponent $\alpha =\min \{1,\frac{n}{m}\}$ is optimal.