Local generalizations of derivatives

TL;DR

This paper explores generalized derivatives based on modulus of continuity, extending classical differentiation concepts to continuous functions that may not be absolutely continuous, with applications to differentiation theorems.

Contribution

It introduces a new class of local derivatives defined via modulus of continuity, broadening the scope of differentiation for continuous functions.

Findings

Generalized derivatives are suitable for continuous functions not necessarily absolutely continuous.

Conditions for the continuity of these generalized derivatives are established.

A generalization of the Lebesgue monotone differentiation theorem is provided.

Abstract

From physical perspective, derivatives can be viewed as mathematical idealizations of the linear growth. The linear growth condition has special properties, which make it preferred. The manuscript investigates the general properties of the local generalizations of derivatives assuming the usual topology of the real line. The concept of derivative is generalized in terms of the class of the modulus of continuity of the primitive function. This definition is suitable for applications involving continuous but possibly non-absolutely continuous functions of a real variable. The main application of the approach is the generalization of the Lebesgue monotone differentiation theorem. On the second place, the conditions of continuity of generalized derivative are also demonstrated.