Analytic functions whose all derivatives keep $\mathbb{Z}$ invariant Is there a real analytic function $f:\mathbb{R} \to \mathbb{R}$ which is not a polynomial function but all its derivatives $f^{(n)}$ satisfy $f^{(n)}(\mathbb{Z})\subseteq \mathbb{Z}$? This question was included in a more flexible question as follows but according to its answers and comments, I realize that the smooth version of this question was obvious. Smooth or analytic functions which keep $\mathbb{Z}$ invariant

Functions satisfying $f^{(n)}(\mathbb Z)\subseteq\mathbb Z$ are called utterly integer valued. There is a simple construction of an utterly integer valued 1-periodic entire function given in Sato, Daihachiro. Utterly integer valued entire functions. I. Pacific J. Math. 118 (1985), no. 2, 523--530. I will just give the idea.

Consider a series of the form

$$f(z)=\sum_{n=0}^\infty a_n\sin^n(2\pi z).$$

Formally at least, $f^{(n)}(0) = a_n n!(2\pi)^n + R(n)$ where $R(n)$ depends only on $a_0,\dots,a_{n-1}.$ Each $a_n$ can be chosen inductively such that $f^{(n)}(0)$ is the closest non-zero integer to $R(n),$ giving $|a_n|\leq 1/n!(2\pi)^n.$ This rate of growth guarantees that the series defines an entire function, which by construction is utterly integer valued.