On sets where $\operatorname{lip} f$ is finite
Volume 249 / 2019
Studia Mathematica 249 (2019), 33-58
MSC: Primary 26A21; Secondary 26A99.
DOI: 10.4064/sm170820-26-5
Published online: 26 April 2019
Abstract
Given a function $f\colon \mathbb{R}\to \mathbb{R}$, the so-called “little lip” function $\operatorname{lip} f$ is defined as follows: $$ \operatorname{lip} f(x)=\liminf_{r\searrow 0}\sup_{|x-y|\le r} \frac{|{f(y)-f(x)}|}{r}. $$ We show that if $f$ is continuous on $\mathbb{R}$, then the set where $\operatorname{lip} f$ is infinite is a countable union of countable intersections of closed sets (that is, an $F_{\sigma \delta}$ set). On the other hand, given a countable union $E$ of closed sets, we construct a continuous function $f$ such that $\operatorname{lip} f$ is infinite exactly on $E$. A further result is that, for a typical continuous function $f$ on the real line, $\operatorname{lip} f$ vanishes almost everywhere.
