A functional equation of Abel revisited (Q1341258)

The equation \(g(x) + g(y) = h(xf(y) + yf(x))\) \((f,g,h\) unknown) was solved by \textit{N. H. Abel} [J. Reine Angew. Math. 21, 386-394 (1827)] under differentiability suppositions. It also belonged to those functional equations which D. Hilbert in 1900, in the second part of the fifth of his famous unsolved problems, proposed for solution under weaker conditions. The author [Aequationes Math. 39, No. 1, 19-39 (1990; Zbl 0694.39004)] determined the continuous solutions on real intervals containing 0. In the present paper the continuous solutions are offered on intervals not containing 0 but with (real) codomain containing 0. Some further reduction of assumptions is also mentioned.