I asked this question on [MSE][1] about 5 months ago, but, even after offering a bounty, I didn't receive any answer, I hope this question isn't too easy for MO.

If we have a set of points $(x_i,y_i)$ with rational coordinates and distinct $x$-coordinates, such that for every (if any) $x_i\in\Bbb{N}$ the corresponding $y_i$ is also an integer, is it always possible to build an integer valued polynomial with rational coefficents passing through those points?


  [1]: https://math.stackexchange.com/questions/732854/integer-valued-polynomial-through-some-known-points