Extreme values of some continuous nowhere differentiable functions

We consider the functions T-n(x) defined as the nth partial derivative of Lebesgue's singular function L-a(x) with respect to a at a = 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.