Characterization of lip sets

Abstract

We denote the local ``little" Lipschitz constant of a function $f:\mathbb{R}\to \mathbb{R}$ by $\mathrm{lip}f$. In this paper we settle the following question: For which sets $E\subset \mathbb{R}$ is it possible to find a continuous function $f$ such that $\mathrm{lip}f=1_E$? In an earlier paper we introduced the concept of strongly one-sided dense sets. Our main result characterizes $\mathrm{lip}1$ sets as countable unions of closed sets which are strongly one-sided dense. We also show that a stronger statement is not true i.e. there are strongly one-sided dense $F_{\sigma }$ sets which are not $\mathrm{lip}1$.