On global extrema of power Takagi functions

By construction, power Takagi functions S_p are similar to Takagi's continuous nowhere differentiable function described in 1903. These real-valued functions S_p have one real parameter p>0. They are defined on the real axis RR"> by the series S_p(x)=∑^∞_n=0 (S₀(2ⁿx)/2ⁿ)^p, where S₀(x) is the distance from real number x to the nearest integer number. We show that for every p>0, the functions S_p are everywhere continuous, but nowhere differentiable on R. Next, we derive functional equations for Takagi power functions. With these, it is possible, in particular, to calculate the values S_p(x) at rational points x. In addition, for all values of the parameter p from the interval (0; 1), we find the global extrema of the functions S_p, as well as the points where they are reached. It turns out that the global maximum of S_p equals to 2^p/(3^p(2^p-1)) and is reached only at points q+1/3 and q+2/3, where q is an arbitrary integer. The global minimum of the functions S_p equals to 0 and is reached only at integer points. Using the results on global extremes, we obtain two-sided estimates for the functions S_p and find the points at which these estimates are reached.