Big and little Lipschitz one sets

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