# Remarks on the Cauchy functional equation and variations of it - Mathematics > Classical Analysis and ODEs

Remarks on the Cauchy functional equation and variations of it - Mathematics > Classical Analysis and ODEs - Download this document for free, or read online. Document in PDF available to download.

Abstract: This paper examines various aspects related to the Cauchy functional equation$fx+y=fx+fy$, a fundamental equation in the theory of functionalequations. In particular, it considers its solvability and its stabilityrelative to subsets of multi-dimensional Euclidean spaces and tori. Several newtypes of regularity conditions are introduced, such as a one in which a complexexponent of the unknown function is locally measurable. An initial valueapproach to analyzing this equation is considered too and it yields a fewby-products, such as the existence of a non-constant real function having anuncountable set of periods which are linearly independent over the rationals.The analysis is extended to related equations such as the Jensen equation, themultiplicative Cauchy equation, and the Pexider equation. The paper alsoincludes a rather comprehensive survey of the history of the Cauchy equation.

Author: ** Daniel Reem**

Source: https://arxiv.org/