Almost orthogonally additive functions
Suppose \((E, \langle . , . \rangle)\) is a real inner product space, \(\text{dim}E\geq 2\) and \((G,+)\) is an abelian group. A function \(f:E\to G\) is called orthogonally additive if it satisfies the equation \(f(x+y)=f(x)+f(y)\) for each \[ (x, y)\in \perp :=\{ (x, y)\in E^2:\langle x, y \rangle=0\}. \] \textit{J. Rätz} [Aequationes Math. 28, 35--49 (1985; Zbl 0569.39006)] proved that such a function has the form \(f(x)=a(\|x\|^2)+b(x)\), with some additive mappings \(a:\mathbb{R} \to G, b:E\to G\), where \(G\) is uniquely 2-divisible. If \(Z\) is ``negligible'' subset of the \((2n-1)\)-dimensional manifold \(\perp \subset \mathbb{R}^{2n}\) and if \(f: \mathbb{R}^{n} \to G\) is ``almost'' orthogonally additive for each \((x, y)\in \perp -Z\), then \(f\) coincides almost everywhere with some othogonally additive mapping.
scientific article; zbMATH DE number 6138098