Properties of functions with monotone graphs

TL;DR

This paper explores the properties of continuous functions whose graphs are monotone sets in a metric space, revealing their differentiability characteristics and Hausdorff dimension.

Contribution

It provides new insights into the differentiability and fractal properties of functions with monotone graphs in metric spaces.

Findings

Functions can be almost nowhere differentiable

Functions are differentiable on a dense set

Hausdorff dimension of the graph is 1

Abstract

A metric space (X,d) is monotone if there is a linear order < on X and a constant c>0 such that d(x,y) < c d(x,z) for all x<y<z in X. Properties of continuous functions with monotone graph (considered as a planar set) are investigated. It is shown, e.g., that such a function can be almost nowhere differentiable, but must be differentiable at a dense set, and that Hausdorff dimension of the graph of such a function is 1.