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 $(\uu, \w)$--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 $x + \uu (h)$, $x - \w (h)$, functions $\uu(h)$, $\w(h)$ are infinitely small at zero, such that for all $h$ from it the inequalities $\uu(h) \neq -\w(h)$, $\uu(h)\cdot \w(h)\geq 0$ 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.