  Let K be a subfield of the real field, D be a discrete subset of K and f :
D^n -> K be a function such that f(D^n) is somewhere dense. Then (K,f) defines
the set of integers. We present several applications of this result. We show
that K expanded by predicates for different cyclic multiplicative subgroups
defines the set of integers. Moreover, we prove that every definably complete
expansion of a subfield of the real field satisfies an analogue of the Baire
Category Theorem.
