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On additive functions fulfilling some additional condition. (English) Zbl 1046.39019

The authors give a positive answer to a generalization of a question asked by G. Szabó [Aequationes Math. 46, 294 (1993)] at the 30th International Symposium on Functional Equations (Oberwolfach, 1992): Let \(D\) be a subset of positive Lebesgue measure of \(\mathbb{R}^{2n}.\) Then any additive function \(f:\mathbb{R}^n\to\mathbb{R}\) satisfying \((x,y)\in D\Rightarrow f(x)f(y)=0\) has to be identically 0. Further generalizations are also offered.

MSC:

39B22 Functional equations for real functions
39B55 Orthogonal additivity and other conditional functional equations
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