Nowhere-monotone differentiable functions and bounded variation

One of the simplest constructions of a differentiable monster-a function from a nontrivial interval J subset of 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 -> Rcontains many restrictions that admit nice representation: for every non-empty open U subset of J, there is a non-trivial interval I subset of U such that f vertical bar(I) is a difference of two increasing differentiable maps, each with its derivative vanishing on a dense subset of I. (c) 2020 Elsevier Inc. All rights reserved.

Automated summary

This paper discusses the difference representation of differentiable functions, proves the existence of differentiable functions that do not have bounded variation, and points out that every differentiable function contains restrictions that can be nicely represented with a difference of increasing differentiable functions.