On Cauchy’s functional equation

Abstract

In this note we shall solve two open problems1 concerning Cauchy's functional equation (C) fix + y) = fix) + fiy)for real-valued functions of a real variable.P. Erdös [2] asked, after learning about a preliminary result of S. Hartman [3], whether one obtains all functions satisfying (C) for almost all pairs (x, y) by simply redefining the functions satisfying (C) for all (x, y) in an arbitrary manner on sets of measure zero.This turns out to be the case (Theorem I), in particular, if f satisfies (C) for almost all (x, y) and is also measurable or only bounded from below on a set of positive measure, then fix) = ex holds almost everywhere as a consequence of well-known theorems of A. Ostrowski [5], H. Kestelman [4].1. Halperin asked whether (C) for all (x, y) in conjunction with

Locations

• Proceedings of the American Mathematical Society - View - PDF