A conditional equation for additive functions

Abstract

Let S denote the set of all pairs (x, y) of real numbers that satisfy the condition x2 + y2 = 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.