## Real Analysis Exchange

### Stability of the Cauchy functional equation over p-adic fields.

#### Abstract

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 $\mathbb{Q}_p$.

#### Article information

Source
Real Anal. Exchange, Volume 31, Number 1 (2005), 125-132.

Dates
First available in Project Euclid: 5 June 2006

Permanent link to this document
https://projecteuclid.org/euclid.rae/1149516796

Mathematical Reviews number (MathSciNet)
MR2218193

Zentralblatt MATH identifier
1099.39019

#### Citation

Arriola, L. M.; Beyer, W. A. Stability of the Cauchy functional equation over p -adic fields. Real Anal. Exchange 31 (2005), no. 1, 125--132. https://projecteuclid.org/euclid.rae/1149516796

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