Characterization of derivations through their actions on certain elementary functions

TL;DR

This paper characterizes real derivations by examining their behavior on elementary functions, establishing conditions under which additive functions are sums of derivations and linear functions.

Contribution

It provides new characterization theorems for real derivations based on their actions on differentiable functions and regularity conditions.

Findings

Additive functions satisfying certain regularity conditions are sums of derivations and linear functions.

Characterization theorems connect derivations with their behavior on elementary functions.

Regularity conditions like measurability or continuity imply specific structural forms.

Abstract

The main aim of this note is to provide characterization theorems concerning real derivations. Among others the following implication will be verified: Assume that $\xi\colon \mathbb{R}\to \mathbb{R}$ is a given differentiable function and for the additive function $d\colon \mathbb{R}\to \mathbb{R}$, the mapping \[ x\longmapsto d(\xi(x))-\xi'(x)d(x) \] is regular (e. g. measurable, continuous, locally bounded). Then $d$ is a sum of a derivation and a linear function.