Self-Affine Singular and Nowhere Monotone Functions Related to the Q-Representation of Real Numbers

Abstract

We study functional, differential, integral, self-affine, and fractal properties of continuous functions belonging to a finite-parameter family of functions with a continuum set of "peculiarities". Almost all functions of this family are singular (their derivative is equal to zero almost everywhere in the sense of Lebesgue) or nowhere monotone, in particular, nondifferentiable. We consider different approaches to the definition of these functions (using a system of functional equations, projectors of symbols of different representations, distribution of random variables, etc.).