Locally Integer Polynomial Functions

The goal of this note is to bring attention to an interesting family of rings: the rings of $\mathbb Z$-valued functions on $\mathbb Z$ and, more generally, infinite subsets of $\mathbb Z$ whose restrictions to all finite sets are given by polynomials with integer coefficients. Our interest in these functions was inspired by the work of Sayak Sengupta on iterations of integer polynomials, but they appear to be of independent interest. In particular, they enjoy some properties reminiscent of the properties of complex analytic functions, including forming a sheaf in the cofinite and density one topologies.