Approximate versions of Cauchy's functional equation
type: Item
instance of: scholarly article
title: Approximate versions of Cauchy's functional equation
zbMATH Open document ID: 0824.39007
author: J. Ralph Alexander
author: C. E. Blair
author: Lee A. Rubel
published in: Illinois Journal of Mathematics
publication date: 1995-06-21T00:00:00Z
review text: The authors prove that if \(f,a,b: \mathbb{R}\to \mathbb{R}\) are measurable functions and there is a \(J\in \mathbb{R}\) such that, for every \(\varepsilon> 0\), \(\mu(\{(x, y): | f(x+ y)- a(x)- b(y)- J|\geq \varepsilon\})\) is finite, then, for some \(\gamma\) and \(\beta\), \(f(x)= \gamma x+ \beta\) almost everywhere. It is also shown that if \(f\in L^ 1[0, \alpha]\) for all \(\alpha> 0\) and, for almost all \(x\), \(\lim_{u\to \infty} {1\over u} \int^ u_ 0 (f(x+ y)- f(x)- f(y)) dy= 0\), then for some \(\gamma\), \(f(x)= \gamma x\) for almost all \(x\geq 0\).
Mathematics Subject Classification ID: 39B22
Mathematics Subject Classification ID: 39B72
zbMATH DE Number: 766573
zbMATH Keywords: Cauchy's functional equation
zbMATH Keywords: Hyers-Ulam stability
reviewed by: Kazimierz Nikodem
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 766573