Big and little Lipschitz one sets

Abstract

Given a continuous function $f: {\mathbb R}\to {\mathbb R}$ we denote the so-called "big Lip" and "little lip" functions by $ {\mathrm {Lip}} f$ and $ {\mathrm {lip}} f$ respectively}. In this paper we are interested in the following question. Given a set $E {\subset} {\mathbb R}$ is it possible to find a continuous function $f$ such that $ {\mathrm {lip}} f=\mathbf{1}_E$ or $ {\mathrm {Lip}} f=\mathbf{1}_E$? For monotone continuous functions we provide the rather straightforward answer. For arbitrary continuous functions the answer is much more difficult to find. We introduce the concept of uniform density type (UDT) and show that if $E$ is $G_δ$ and UDT then there exists a continuous function $f$ satisfying $ {\mathrm {Lip}} f =\mathbf{1}_E$, that is, $E$ is a $ {\mathrm {Lip}} 1$ set. In the other direction we show that every ${\mathrm {Lip}} 1$ set is $G_δ$ and weakly dense. We also show that the converse of this statement is not true, namely that there exist weakly dense $G_{δ}$ sets which are not $ {\mathrm {Lip}} 1$. We say that a set $E\subset \mathbb{R}$ is ${\mathrm {lip}} 1$ if there is a continuous function $f$ such that ${\mathrm {lip}} f=\mathbf{1}_E$. We introduce the concept of strongly one-sided density and show that every ${\mathrm {lip}} 1$ set is a strongly one-sided dense $F_σ$ set.