This paper is devoted to the development of the theory of locally complex continuous functions with fractal properties, constructed in terms of various coding systems (representations) of real numbers. These include Cantor numeral systems, Q ̃-representations, and an original newly created coding system for real numbers (B-representation) using a bidirectionally infinite alphabet. In it, the structural, variational, topological-metric, integrodifferential, and fractal properties of three classes of continuous functions are studied, including: nowhere monotonic functions (those that do not have monotonic intervals), singular functions (those that are continuous, nonconstant, and have a derivative equal to zero almost everywhere in terms of the Lebesgue measure), as well as functions that do not have monotonic intervals except for intervals of constancy.
The argument of these functions is defined using one of the aforementioned representations, and the function's value is determined by a specific infinite matrix whose column sums equal one.
The most thoroughly studied is the class of functions associated with the newly substantiated and detailed B-representation of numbers within the unit interval. For functions of this class, a formula has been derived for calculating the Lebesgue measure of the set of non-constancy of the function, which is the difference between the domain of definition and the union of intervals of constancy. Necessary and sufficient conditions for its zero-dimensionality have been established. A criterion for the function's membership in the class of singular Cantor-type functions has been substantiated.
It has been proven that a function is nowhere monotonic if and only if there are no zeros among the elements of its defining matrix, and an infinite number of its columns contain negative elements. A formula for calculating the variation of the function has been derived. Necessary and sufficient conditions under which the function has unbounded variation have been found. In particular, it has been shown that it has unbounded variation when all columns of the matrix are identical and contain no zeros. Conditions have been specified under which the function has Cantor or quasi-Cantor type, unbounded variation, and no monotonic intervals other than intervals of constancy.
It has been established that in the case where all columns of the defining matrix are identical and contain no zeros, the graph of the function is an N-self-affine set. The self-affine structure of the function's graph has been used to compute the definite integral.
Under the condition that all columns of a matrix are identical and its elements are positive, sufficient conditions have been found for the function to be a singular strictly increasing probability distribution function on the unit interval.
