This is a follow up of an interesting recent question on the topic. The answer given there by fedia shows that the matter is rich and complicated, and I can't resist to submit here a further question.

Q1.What is the closure of theinteger valued polynomialsin $C^0(\mathbb{R},\mathbb{R})$ w.r.to the uniform convergence on bounded sets?

For instance, a nontrivial element in the closure is, by the Euler infinite product

$$\frac {\sin(\pi x)} \pi x = \lim_{n \to \infty}\, \binom {x-1} n \binom {x+n} n $$

but can one reach all integers-to-integers continuous functions on $\mathbb{R}$?

(After a quick Google search, among the large literature on integer valued polynomials, I was only able to find this book concerning approximation by IVP, that however treats approximation of functions on the p-adic integers rather than on $\mathbb{R}$; I'm not p-adic enough to understand if there are implications to the real case).

[edit] A further question raised by SJR

Q2.What is the closure of thepolynomials with integer coefficientsin $C^0(\mathbb{R},\mathbb{R})$ w.r.to the uniform convergence on bounded sets?

At a first glance I'd have said $\mathbb{Z}[x]$ is closed in $C^0(\mathbb{R}, \mathbb{R})$ for some elementary reason, but now I don't see it. It is certainly a countable, metrizable commutative group; so it is closed iff it is discrete. It has an additional ring structure and it's a monoid by composition, both structure being compatible with the topology.