Nowhere-monotone differentiable functions and set of monstrous shift

Abstract

Given a strictly increasing differentiable function f:R→R whose derivative vanishes on a dense subset of R, it has been shown by Ciesielski that there is at least a comeager amount of t∈R such that f(x)−f(x−t) is nowhere-monotone. In this paper, we show that the amount of their shifted difference that are nowhere-monotone is not only comeager but also of full Lebesgue measure in R. Interestingly, we also provide a specific pair of strictly increasing differentiable functions φ,ψ:R→R such that φ(x)−ψ(x−t) is nowhere-monotone for every t∈R.