2019/05/27 by Buczolich, Zoltán, Hanson, Bruce, Maga, Balázs +1 · 1 citation
#26A16 #28A05 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.1905.11081
Given a continuous function f: \mathbb R→ \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 ⊂ \mathbb R is it possible to find a continuous function f such that \mathrm lip f=1E or \mathrm Lip f=1E? 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 Gδ and UDT then there exists a continuous function f satisfying \mathrm Lip f =1E, that is, E is a \mathrm Lip 1 set. In the other direction we show that every \mathrm Lip 1 set is Gδ and weakly dense. We also show that the converse of this statement is not true, namely that there exist weakly dense Gδ sets which are not \mathrm Lip 1. We say that a set E⊂ ℝ is \mathrm lip 1 if there is a continuous function f such that \mathrm lip f=1E. We introduce the concept of strongly one-sided density and show that every \mathrm lip 1 set is a strongly one-sided dense Fσ set.