Characterization of derivations through their actions on certain elementary functions

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.