On additive functions with additional derivation properties

TL;DR

This paper introduces generalized derivations for smooth vector-valued functions, establishes calculus rules, and proves that functions satisfying certain addition theorems are derivable, extending previous results and posing open problems.

Contribution

It develops a new framework for generalized derivations and applies it to extend existing theorems in the theory of additive functions.

Findings

Established calculus rules for generalized derivations.

Proved that functions satisfying specific addition theorems are derivable.

Extended previous results and formulated open problems.

Abstract

The purpose of this paper is to introduce the notion of a generalized derivation which derivates a prescribed family of smooth vector-valued functions of several variables. The basic calculus rules are established and then a result derived which shows that if a function $f$ satisfies an addition theorem whose determining operation is derivable with respect to an additive function $d$, then the function $f$ is itself derivable with respect to $d$. As an application of this approach, new proof of a generalization of a recent result of Maksa is obtained. We also extend the result of Nishiyama and Horinouchi and formulate two open problems.