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

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 $g_λ(x),$ where $λ\in(0,1)$. 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 $λ=\frac12$. All functions of the family are continuous, strictly increasing and map the segment $[0,1]$ onto itself. Moreover, they are singular i.e. $\forall λ$ the derivative $g'_λ(x),$ if exists, can take only two values: 0 and $+\infty.$ In this paper we consider two functions of the class which correspond to $λ$ 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 $g'_λ(x)$ 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.

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