Generalization of the classical derivative and analogue of the differentiation operator as a toolset for studying the differential properties of functions

Abstract

In this paper, we consider a $(u, w)$--derivative that is a generalization of a classical derivative and an operator defined by the equality $Sqxuv f (x) = \lim\limits_{h \rightarrow 0}\frac{Squv f(x)}{Squv x}$, where $Squv f(x)$ is the oscillation of the function $ f $ on the interval with the ends at the points $ x + u (h) $, $ x - w (h) $, functions $u(h)$, $w(h)$ are infinitely small at zero, such that for all $ h $ from it the inequalities $u(h) \neq -w(h)$, $u(h)\cdot w(h)\geq 0$ holds for a punctured neighbourhood of zero.
Their properties and relationships with the classical derivative are described. Their application for the derivation of differential properties is presented by the example of a model class of nowhere monotone functions.