Alternative Cauchy equation in three unknown functions (Q2055264)

The author studies the following Cauchy equation for two unknown functions \(f,g: \mathbb{R}\to \mathbb{R}\): \[ [f(x+y)-f(x)-f(y)] [g(x+y)-g(x)-g(y)] = 0, \] and the equation for three unknown functions \(f,g,h: \mathbb{R}\to \mathbb{R}\): \[ [f(x+y)-f(x)-f(y)] [g(x+y)-g(x)-g(y)] [h(x+y)-h(x)-h(y)] = 0. \] For the first case it is proved that a pair \((f,g)\) of continuously differentiable functions is a solution if and only if at least one of the functions is linear. For the second case the author provides examples to illustrate the existence of nontrivial \(\mathcal{C}^\infty\) solutions. Under the assumption of real analytic functions, the author shows that the solutions must be linear.