Short remark to the Rimán's Theorem

In this paper the general solution of the functional equation f (x+ y) = g(x) + h(y) ((x, y) is an element of D) is given with unknown functions f : Dx+y -> Y, g: Dx -> Y, h: Dy -> Y where D subset of G2 is a nonempty, open set, (G, <=) is an ordered, dense, Abelian group, the topology on G is generated by the open intervals of G, the sets Dx, Dy, Dx+y are defined by Dx := {u is an element of G | there exists v is an element of G : (u, v) is an element of D}, Dy := {v is an element of G | there exists u is an element of G : (u, v) is an element of D}, Dx+y:= {z is an element of G | there exists(u, v) is an element of D : z = u + v}, and Y (+) is an Abelian group. The main result of the article is a common generalization of similar results by L. Szekelyhidi and J. Riman. Analogous theorem concerning logarithmic functions is also shown.