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=\mathbf{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$.