On sets where $\mathrm{lip}f$ is finite

Abstract

Given a function $f:\mathbb{R}\to \mathbb{R}$, the so-called "little lip" function $\mathrm{lip}f$ is defined as follows: \begin{equation*} \operatorname{lip} f(x)=\liminf_{r{\scriptscriptstyle \searrow} 0}\sup_{|x-y|\le r} \frac{|f(y)-f(x)|}{r}. \end{equation*} We show that if $f$ is continuous on $\mathbb{R}$, then the set where $\mathrm{lip}f$ is infinite is a countable union of a countable intersection of closed sets (that is an $F_{\sigma \delta }$ set). On the other hand, given a countable union of closed sets $E$, we construct a continuous function $f$ such that $\mathrm{lip}f$ is infinite exactly on $E$. A further result is that for the typical continuous function $f$ on the real line $\mathrm{lip}f$ vanishes almost everywhere.

arXiv comment27 pages, 3 figures. We updated affiliations and the acknowledgements section and reformatted the paper. It is accepted for publication in Studia Mathematica