Derivations and linear functions along rational functions

TL;DR

This paper characterizes derivations and linear functions through properties of rational functions, establishing conditions under which additive functions can be expressed as sums of derivations and linear functions.

Contribution

It provides new characterization theorems for derivations and linear functions based on regularity conditions of a specific rational function involving additive functions.

Findings

Additive functions can be represented as sums of derivations and linear functions under certain regularity conditions.

The paper establishes conditions ensuring linearity of additive functions.

Characterization theorems are proved for derivations and linear functions using rational functions.

Abstract

The main purpose of this paper is to give characterization theorems on derivations as well as on linear functions. Among others the following problem will be investigated: Let $n\in\mathbb{Z}$, $f, g\colon\mathbb{R}\to\mathbb{R}$ be additive functions, $<{array}{cc} a&b c&d {array}>\in\mathbf{GL}_{2}(\mathbb{Q})$ be arbitrarily fixed, and let us assume that the mapping \[ \phi(x)=g<\frac{ax^{n}+b}{cx^{n}+d}>-\frac{x^{n-1}f(x)}{(cx^{n}+d)^{2}} \quad <x\in\mathbb{R}, cx^{n}+d\neq 0> \] satisfies some regularity on its domain (e.g. (locally) boundedness, continuity, measurability). Is it true that in this case the above functions can be represented as a sum of a derivation and a linear function? Analogous statements ensuring linearity will also be presented.