Osaulenko R. Singular non where monotonic functions and its fracral properties

The work is performed in the field of constructive theory of locally complicated continuous functions with fractal properties. It is dedicated to singular (monotonic and nowhere monotonic) functions and ways of their study. To study the that functions, the constructions of differential operators are proposed: (u,v)-derivative and analogue of it, logarithmic (u,v)- derivative and analogue of it. The properties of these concepts are studied, their connections are established; the new classes of nowhere monotone (singular and nondifferentiable) functions (bounded and unbounded variation) their capacity and potency are demonstrated. In particular, the application of the proposed differentiation operators with their adaptations in the theories of fractals, analysis, ordinary differential equations, series and improper integrals is shown.