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

TL;DR

This paper characterizes the structure of the set where the little lip function is infinite for continuous functions on the real line, showing it is an $F_{\sigma\delta}$ set, and constructs functions with prescribed sets of infinite little lip.

Contribution

It provides a detailed topological description of the set where the little lip function is infinite and constructs continuous functions with specific sets of infinite little lip.

Findings

The set where $ ext{lip} f$ is infinite is an $F_{\sigma\delta}$ set for continuous $f$.

Any countable union of closed sets can be realized as the set where $ ext{lip} f$ is infinite for some continuous $f$.

For a typical continuous function, $ ext{lip} f$ vanishes almost everywhere.

Abstract

Given a function $f\colon \mathbb{R}\to \mathbb{R}$, the so-called "little lip" function $\operatorname{lip} f$ 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 $\operatorname{lip} f$ 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 $\operatorname{lip} f$ is infinite exactly on $E$. A further result is that for the typical continuous function $f$ on the real line $\operatorname{lip} f$ vanishes almost everywhere.