Big and little Lipschitz one sets

Abstract

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