On sets where lip f is finite

2017/08/28 by Buczolich, Zoltán, Hanson, Bruce, Rmoutil, Martin +1 · 1 citation
#26A21 #26A99 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1708.08220

Abstract

Given a function f\colon ℝ→ ℝ, the so-called "little lip" function lip f is defined as follows: lip f(x)=\liminf_r\scriptscriptstyle \searrow 0sup_{|x-y|≤ r} (|f(y)-f(x)|)/(r). We show that if f is continuous on ℝ, then the set where lip f is infinite is a countable union of a countable intersection of closed sets (that is an F_σδ set). On the other hand, given a countable union of closed sets E, we construct a continuous function f such that lip f is infinite exactly on E. A further result is that for the typical continuous function f on the real line lip f vanishes almost everywhere.

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