On a functional equation of Bruce Ebanks

In the paper Brillouet-Belluot and Ebanks (Aequationes Math 60:233-242, 2000), the authors found all continuous functions f: [0, 1] -> [0, + infinity) which verify f(0) = f(1) = 0 and the functional equation f (xy + cf (x) f (y)) = xf (y) + yf(x) + d f(x)f(y) (1) where c and d are given real numbers with c not equal 0. In the present paper we obtain all continuous solutions f : R -> R of the functional equation (1).