Kalashnikov A. Singular and nowhere monotonic functions as solutions of systems of functional equations

The thesis is devoted to continuous on functions with complicated local structure (singular, nowhere monotonic and nowhere differentiable functions) being solutions of some class of systems of functional equations. Using the analysis of structural, self-similar and self-affine properties of known strictly increasing singular functions (Salem, Salem-Takбcs and Minkowski functions) and functional relations which they satisfy we give generalizations and study their extremal, differential, integral and fractal properties. All functions under consideration are well defined as continuous solutions of systems of functional equations. Transition to the -representation of real numbers generalizing the -adic representation do not lead to loss of continuity, and use of alternating series enriches the family of functions with singular non-monotonic, nowhere monotonic and non-differentiable functions. Fractal properties of functions described in the thesis (in particular, self-affinity of their graphs as a consequence of self-similarity of the -representation's geometry) help to study local and global properties of functions and calculate the Lebesgue integrals. In the work we propose a one-parameter generalization of singular Minkowski function defined in terms of regular continued fractions and alternating binary series (special encoding of real numbers with infinite alphabet). We prove that every representative of this family is strictly increasing singular function that is continuous function such that its derivative is equal to zero Lebesgue almost everywhere.