Characterization of the Local Growth of Two Cantor-Type Functions

Dimiter Prodanov

doi:10.3390/fractalfract3030045

Abstract

The Cantor set and its homonymous function have been frequently utilized as examples for various physical phenomena occurring on discontinuous sets. This article characterizes the local growth of the Cantor’s singular function by means of its fractional velocity. It is demonstrated that the Cantor function has finite one-sided velocities, which are non-zero of the set of change of the function. In addition, a related singular function based on the Smith–Volterra–Cantor set is constructed. Its growth is characterized by one-sided derivatives. It is demonstrated that the continuity set of its derivative has a positive Lebesgue measure of 1/2.

Highlights

The Cantor set and its homonymous function have been frequently utilized as examples for various physical phenomena occurring on discontinuous sets
A related singular function based on the Smith–Volterra–Cantor set is constructed
The Cantor set is an example of a perfect set that is at the same time nowhere dense

Summary

Introduction

The Cantor set is an example of a perfect set (i.e., closed and having no isolated points) that is at the same time nowhere dense. The Cantor function is the standard example of a singular function whose derivative vanishes almost everywhere in the unit interval. The function is the unique solution of Equation (1) in the class of all bounded functions f : [0; 1] → R ([7,8] (Proposition 10.6.2)) It has fixed points F(0) = 0 and F(1) = 1, among others. The Cantor function can be defined as the map between the Cantor ternary set C and the set of dyadic rationals D \ {1/2} extended by continuity on the entire unit interval. This approach will be used for the second example presented in the paper

Bounds of Growth of the Cantor’S Function

Point-Wise Oscillation of Functions

Fractional Velocity

The Smith–Volterra–Cantor Set and Its Related Singular Function