. Finally, Suppose that, for some a, f (a) < a. Since a divides f (a) ? f (0) ? f (a) < a, we have f (a) = f (0) hence f is constant on {0

J. Myhill, J. Pin, and P. V. Silva, On profinite uniform structures defined by varieties of finite monoids, Pin and P.V. Silva, On uniformly continuous functions for some profinite topologies, pp.112-137, 1957.

R. Stansifer, Presburgers Article on Integer Arithmetic: Remarks and Translation, Cornell tech reports, pp.84-639, 1813.