Remarks on the Cauchy functional equation and variations of it

TL;DR

This paper explores the solvability, stability, and regularity conditions of the Cauchy functional equation and related equations, introducing new concepts and providing a comprehensive historical survey.

Contribution

It introduces new regularity conditions, such as local measurability of complex exponents, and extends analysis to related functional equations, offering novel insights and methods.

Findings

Existence of a non-constant real function with uncountably many linearly independent periods

New regularity conditions for solutions, including local measurability of complex exponents

Extended analysis to Jensen, multiplicative Cauchy, and Pexider equations

Abstract

This paper examines various aspects related to the Cauchy functional equation $f(x+y)=f(x)+f(y)$, a fundamental equation in the theory of functional equations. In particular, it considers its solvability and its stability relative to subsets of multi-dimensional Euclidean spaces and tori. Several new types of regularity conditions are introduced, such as a one in which a complex exponent of the unknown function is locally measurable. An initial value approach to analyzing this equation is considered too and it yields a few by-products, such as the existence of a non-constant real function having an uncountable set of periods which are linearly independent over the rationals. The analysis is extended to related equations such as the Jensen equation, the multiplicative Cauchy equation, and the Pexider equation. The paper also includes a rather comprehensive survey of the history of the Cauchy equation.