One class of continuous functions related to Engel series and having complicated local properties

TL;DR

This paper introduces a new class of continuous functions defined via Engel series, exhibiting complex local behaviors such as singularity, nowhere monotonicity, and non-differentiability, with detailed analysis of their properties.

Contribution

It constructs a novel class of continuous functions with intricate local properties using Engel series, expanding understanding of functions with complex singularities.

Findings

Functions have a continuum of peculiarities including singularities and non-differentiability.

The functions exhibit complex fractal and extremal properties.

Structural and differential analyses reveal intricate local behaviors.

Abstract

We construct and study the class of continuous on $[0, 1]$ functions with continuum set of peculiarities (singular, nowhere monotonic, and non-differentiable functions are among them). The representative of this class is the function defined by the Engel representation of argument and some convergent real series. We study local and global properties of this function: structural, extremal, differential, integral, and fractal properties.