Integer-valued definable functions

We present a dichotomy, in terms of growth at infinity, of analytic functions definable in the real exponential field which take integer values at natural number inputs. Using a result concerning the density of rational points on curves definable in this structure, we show that if a definable, analytic function f : [0,∞)k → |R is such that f(|N^k) ⊆ |Z, then either sup|¯x| |f(¯x)| grows fasterthan exp(rδ), for some δ > 0, or f is a polynomial over Q