Characterization of derivations through their actions on certain elementary functions

Eszter Gselmann

Published 2013-07-16Version 1

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.