Non-differentiable functions defined in terms of classical representations of real numbers. (English) Zbl 1447.26003

Serbenyuk, S. O.

J. Math. Phys. Anal. Geom. 14, No. 2, 197-213 (2018).

Summary: The present paper is devoted to the functions from a certain subclass of non-differentiable functions. The arguments and values of the considered functions are represented by the \(s\)-adic representation or the nega-\(s\)-adic representation of real numbers. The technique of modeling these functions is the simplest as compared with the well-known techniques of modeling non-differentiable functions. In other words, the values of these functions are obtained from the \(s\)-adic or nega-\(s\)-adic representation of the argument by a certain change of digits or combinations of digits. Integral, fractal and other properties of the functions are described.

MSC:

26A27	Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives
11B34	Representation functions
11K55	Metric theory of other algorithms and expansions; measure and Hausdorff dimension
39B22	Functional equations for real functions

Keywords:

nowhere differentiable function; \(s\)-adic representation; nega-\(s\)-adic representation; non-monotonic function; Hausdorff-Besicovitch dimension