Continuity of functions operating on characteristic functions

For a locally compact abelian group G let P-0(C) denote the set of all characteristic functions on G (i.e., continuous positive definite functions phi with phi(0) = 1). A complex-valued function f is said to operate on P-0(G) if f(phi((.))) is an element of P-0(G) whenever phi is an element of P-0(G). The natural domain for functions operating on P-0(G) is the set D(G) = {z is an element of C: z = phi(g), g is an element of G, phi is an element of P-0(G)}. It is known that every function operating on P-0(G) is continuous on the interior of D(C) for each infinite group G. On the other hand, functions discontinuous on D(G) operate on P-0(C) for any discrete G. We show that, for any non-discrete group G, every function operating on P-0(G) is continuous on D(G). Together with some earlier results, this statement allows us to obtain a constructive description of operating functions on the whole set D(G).