Nowhere-monotone differentiable functions and bounded variation

Abstract

One of the simplest constructions of a differentiable monster—a function from a nontrivial interval J⊆R into R that is everywhere differentiable but monotone on no interval—is as a difference of two strictly increasing differentiable functions, each with its derivative vanishing on a dense subset of its domain. The goal of this work is to characterize differentiable monsters that can be represented in such a “nice” way as those that are a difference of two increasing everywhere differentiable functions. We show that there are differentiable monsters that are not of bounded variation, so they clearly do not admit such “nice” representation. On the other hand, every differentiable monster f:J→R contains many restrictions that admit “nice” representation: for every non-empty open U⊆J, there is a non-trivial interval I⊆U such that f↾I is a difference of two increasing differentiable maps, each with its derivative vanishing on a dense subset of I.