Summary.
Let S denote the set of all pairs (x, y) of real numbers that satisfy the condition x 2 + y 2 = 1. It was asked by Walter Benz in 1988 whether an additive real function f that fulfils the equation x f (x) + y f (y) = 0 or y f (x) = x f (y) for every (x, y) ∈ S must be a derivation or a linear function, respectively. Both questions are answered in the affirmative.
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Manuscript received: November 30, 2004.
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Boros, Z., Erdei, P. A conditional equation for additive functions. Aequ. math. 70, 309–313 (2005). https://doi.org/10.1007/s00010-005-2810-x
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DOI: https://doi.org/10.1007/s00010-005-2810-x