Functions consistent with real numbers, and global extrema of functions in exponential Takagi class

Abstract: The functions of the Takagi exponential class are similar in construction to the continuous, nowhere differentiable Takagi function described in 1901. They have one real parameter $v\in (-1;1)$ and at points $x\in{\mathbb R}$ are defined by the series $T_v(x) = \sum_{n=0}^\infty v^n T_0(2^nx)$, where $T_0(x)$ is the distance between $x$ and the nearest integer point. If $v=1/2$ then $T_v$ coincides with Takagi's function. In this paper, for different values of the parameter $v$, we study the global extremes of the functions $T_v$, as well as the sets of extreme points. All functions of $T_v$ have a period of $1$, so they are investigated only on the segment $[0;1]$. This study is based on the properties of consistent and anti-consistent polynomials and series, which the first half of the work is devoted to.