Utility of extension of functional equations - when possible

摘要

Jean-Claude Falmagne observed in 1981 [On a recurrent misuse of classical functional equation result. Journal of Mathematical Psychology, 23, 190-193] that, even under regularity assumptions, not all solutions of the functional equation k(s + t) = k(s) + k(t), important in many fields, also in the theory of choice, are of the form k(s) = Cs. This is certainly so when the domain of the equation (the set of (s, t) for which the equation is satisfied) is finite. We mention an example showing that this can happen even on some infinite, open, connected sets (open regions). The more general equations k(s + t) = t(s) + n(t) and k(s + t) = in(s)n(t), called Pexider equations, have been completely solved on R-2. In case they are assumed valid only on an open region, they have been extended to R-2 and solved that way (the latter if k is not constant). In this paper their common generalization