The continuous solution of a functional equation of Abel
In 1827, \textit{N. H. Abel} [J. Reine Angew. Math. 2, 386-394] obtained the differentiable solutions of the functional equation \(\phi (x)+\phi (y)=\psi (xf(y)+yf(x))\) on the reals. In 1900, Hilbert asked if this can be solved under weaker conditions than differentiability. In 1987, \textit{J. Aczél} drew attention to this question again and asked for its continuous solutions [Bull. Am. Math. Soc., New. Ser. 20, No.2, 153-163 (1989; Zbl 0676.39004)]. In this paper, the author nicely solves the equation under the assumptions that the functions are continuous, and that the equation holds for all x, y in an interval containing 0.
scientific article; zbMATH DE number 4136640