On pointwise Hölder functions.

Abstract

Let $f: X \rightarrow {\mathbb R}$, where $X$ is a subset of ${\mathbb R}^{k}$. We observe that in order for $f$ be almost everywhere pointwise Hölder it is enough that $f$ satisfy the Hölder condition inside angles of a fixed width. In this analysis, density points of $X$ play a primary role. This has some interesting consequences concerning summability of a naturally defined coefficient.