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

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.