Multiplicative derivations on C(X)

Let X be a completely regular topological space satisfying the first axiom of countability with no isolated points, and let C(X) be the algebra of all continuous functions on X. A mapping d:C(X) --> C(X) is called a multiplicative derivation if d(fg) = fd(g) + gd(f) for every pair of functions f,g is an element of C(X) (no linearity or continuity of d is assumed). We obtain a complete description of such mappings and give examples to show that the above assumptions on the space X are essential.