On global extrema of power Takagi functions

Abstract

By construction, power Takagi functions Sp are similar to Takagi's continuous nowhere differentiable function described in 1903. These real-valued functions Sp(x) have one real parameter p>0 . They are defined on the real axis R by the series Sp(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 Sp 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 Sp(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 Sp , as well as the points where they are reached. It turns out that the global maximum of Sp equals to 2p/(3p(2p−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 Sp equals to 0 and is reached only at integer points. Using the results on global extremes, we obtain two-sided estimates for the functions Sp and find the points at which these estimates are reached.