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2005-2006 Stability of the Cauchy functional equation over p-adic fields.
L. M. Arriola, W. A. Beyer
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Real Anal. Exchange 31(1): 125-132 (2005-2006).


This paper corrects some errors found in [2], which discusses an extension of Lorentz transformations over a non-Archimedean valued field; namely, the $p$-adic field $\mathbb{Q}_{p}$. The paper [2] is based on the results given by Hyers [7] which showed that for a continuous function $f$ defined on $\mathbb{R}$, the Cauchy functional equation $f(x + y) = f(x) + f(y)$ is stable. By stable we mean that if there exists $\epsilon >0$ such that $\| f(x + y) - f(x) - f(y) \| < \epsilon$, $\forall x,y$, then there exists a unique and continuous $\mathcal{L}$ such that $\| \mathcal{L}(x) - f(x) \| \leq \epsilon$, $\forall x$ and $\mathcal{L}(x + y) = \mathcal{L}(x) + \mathcal{L}(y)$. In this paper, we show this result is true on the $p$-adic field $\mathbb{Q}_p$.


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L. M. Arriola. W. A. Beyer. "Stability of the Cauchy functional equation over p-adic fields.." Real Anal. Exchange 31 (1) 125 - 132, 2005-2006.


Published: 2005-2006
First available in Project Euclid: 5 June 2006

zbMATH: 1099.39019
MathSciNet: MR2218193

Primary: 11S80 , 39B22

Keywords: Cauchy functional equation , p-adic field , stability

Rights: Copyright © 2005 Michigan State University Press

Vol.31 • No. 1 • 2005-2006
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