H\"older continuity and differentiability on subsequences

TL;DR

The paper demonstrates that any function from a subset of Euclidean space to another can be restricted to a subset where it exhibits H"older continuity at almost every point, and differentiability can be achieved when the domain dimension exceeds or equals the codomain dimension.

Contribution

It establishes that functions can be restricted to subsets with limit points to attain H"older continuity and differentiability, with optimal H"older exponent characterization.

Findings

Functions become $C^{0,eta}$-continuous on certain subsets at almost every point.

Differentiability can be achieved when the domain dimension is at least the codomain dimension.

The H"older exponent $rac{n}{m}$ is proven to be optimal.

Abstract

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