The functional equation \(f(\mathcal{A}+\mathcal{B})=g(\mathcal{A})+h(\mathcal{B})\)

Abstract. We shall investigate the functional equation \(f(a+b)=g(a)+h(b)\) for all \(a\in{\mathcal A}, b\in{\mathcal B}\), where \({\mathcal A}\) and \({\mathcal B}\) are subsets of natural numbers satisfying some condition.