Derivatives of two functions belonging to the Denjoy–Tichy–Uits family

Abstract:

The family of singular functions $g_{\lambda }(x)$, where $\lambda \in (0,1)$, was first considered by Denjoy in 1938 and was rediscovered by Tichy and Uits in 1995. Among these functions, the Minkowski function $?(x)$, which corresponds to $\lambda =\frac 12$, is most widely known. For singular functions, it is of interest to know conditions on $x$ that ensure that $g’_{\lambda }(x)=0$ or $g’_{\lambda }(x)=\infty$. For the Minkowski function, this problem was first treated in 2001 by Paradis, Viader, and Bibiloni, and was basically solved in 2008 by Moshchevitin, Dushistova, and Kan. In the present paper, the derivatives of the functions $g_{\lambda }(x)$ are studied for the values of the parameter $\lambda$ equal to $\frac {\sqrt 5-1}2$ and $1-\frac {\sqrt 5-1}2$. The constants obtained in the paper are sharp.