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

Characterization of derivations through their actions on certain elementary functions