Extreme values of some continuous nowhere differentiable functions

Abstract

We consider the functions $T_n(x)$ defined as the $n$ th partial derivative of Lebesgue's singular function $L_a(x)$ with respect to $a$ at $a=\frac{1}{2}$ . This sequence includes a multiple of the Takagi function as the case $n=1$ . We show that $T_n$ is continuous but nowhere differentiable for each $n$ , and determine the Holder order of $T_n$ . From this, we derive that the Hausdorff dimension of the graph of $T_n$ is one. Using a formula of Lomnicki and Ulam, we obtain an arithmetic expression for $T_n(x)$ using the binary expansion of $x$ , and use this to find the sets of points where $T_2$ and $T_3$ take on their absolute maximum and minimum values. We show that these sets are topological Cantor sets. In addition, we characterize the sets of local maximum and minimum points of $T_2$ and $T_3$ .