On the derivative of two functions from Denjoy-Tichy-Uitz family

TL;DR

This paper studies the derivatives of specific functions from the Denjoy-Tichy-Uitz family, focusing on conditions under which these derivatives exist and their values, especially for functions related to the golden ratio.

Contribution

It provides new theorems establishing necessary conditions for the existence and values of derivatives of two particular functions in the family, with non-improvable constants.

Findings

Derived conditions for derivative existence at specific points.

Identified the derivative values as 0 or infinity under certain conditions.

Focused on functions associated with special parameters related to the golden ratio.

Abstract

The family of functions, we investigate in this article, was originally introduced by A.Denjoy and later rediscovered by R Tichy and J. Uitz. We denote the functions of the family by $g_{\lambda}(x),$ where $\lambda\in(0,1)$. The definition will be given in the following section. The most famous function of the family is the Minkiowski question-mark function. As we would see, it corresponds to $\lambda=\frac12$. All functions of the family are continuous, strictly increasing and map the segment $[0,1]$ onto itself. Moreover, they are singular i.e. $\forall \lambda$ the derivative $g'_{\lambda}(x),$ if exists, can take only two values: 0 and $+\infty.$ In this paper we consider two functions of the class which correspond to $\lambda$ equals $\frac{\sqrt5-1}2$ or $1-\frac{\sqrt5-1}2.$ The aim of this paper is to prove some theorems about essential conditions on x such that if the condition holds then the derivative $g'_{\lambda}(x)$ exists and has determined value. The constants used in our theorems are non-improvable. Our paper is wirtten in Russian. However Introduction and the formulation of main results are written in English.