Alternative Cauchy equation in three unknown functions

In this paper we deal with the product of two or three Cauchy differences equaled to zero. We show that in the case of two Cauchy differences, the condition of absolute continuity and differentiability of the two functions involved implies that one of them must be linear, i.e., we have a trivial solution. In the case of the product of three Cauchy differences the situation changes drastically: there exists non trivial C∞ solutions, while in the case of real analytic functions we obtain that at least one of the functions involved must be linear. Some open problems are then presented.