A class of nowhere continuous solutions of Baxter's equation (Q1084267)

In response to a question of P. Volkmann and H. Weigel, the author constructs functions f which are solutions of the functional equation \(f(x)f(y)=f(xf(y)+yf(x)-xy)\), \(f: R\to R\), which are respectively (a) discontinuous at all points of R, (b) discontinuous only at the integers, (c) discontinuous only at 0, (d) discontinuous only at 1 and -1.