Redundancy conditions for the functional equation \(f(x+h(x))=f(x)+f(h(x))\) (Q2266392)

The equation appearing in the title is called sometimes conditional Cauchy equation. The problem considered by the author may be stated as follows: under what assumptions imposed on \(h: {\mathbb{R}}\to {\mathbb{R}}\) the only solutions of this equation in a certain class are additive functions (i.e. satisfying the classical Cauchy equation)? Several results exist already in this direction, let us mention \textit{M. Zdun} [Aequationes Math. 8, 229-232 (1972; Zbl 0249.39004)], \textit{J. Dhombres} [1st Franco- southeast Asian math. Conf., Singapore 1979, Spec. Issue of Southeast Asian Bull. Math., 51-71 (1979; Zbl 0421.39005)] and, most generally, the author himself [Boll. Unione Mat. Ital., VI. Ser., B 2, 391-402 (1983; Zbl 0509.39010)]. In the paper under review the author proves that if zeros of h and \(h+id\) are ''rare'' in a certain sense then any continuous and differentiable at zero solution of the considered equation is additive. Some examples are given which show that the assumptions cannot be weakened in general.