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This question is closely related to the thread The closures in $C^0(\mathbb R,\mathbb R)$ of the set of integer valued polynomials, resp, of polynomials with integer coefficients. (Recall that a polynomial $p$ is called integer valued if $p(n)\in\mathbb Z$ for all $n\in\mathbb Z$.)

I would like to know if the closure of integer valued polynomials in $C^0(\mathbb C,\mathbb C)$ is the set of complex continuous functions that take integer values on integers.

The arguments on the already cited thread do not seem to generalize easily.

Thanks in advance for any hint.

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    $\begingroup$ Two trivial remarks: (1) If $p:\mathbb Z\to\mathbb Z$, then $p$ has real coefficients; (2) if $p_n\to f$ locally uniformly, then $f$ is holomorphic. $\endgroup$
    – Christian Remling
    Commented Jun 30, 2014 at 20:14
  • $\begingroup$ Does "polynomial" here mean "holomorphic polynomial", i.e. a polynomial in $z$, or is $\bar{z}$ an integer valued polynomial? $\endgroup$
    – Nate Eldredge
    Commented Jul 1, 2014 at 4:48
  • $\begingroup$ Polynomials means here "polynomials in $z$ only" $\endgroup$
    – joaopa
    Commented Jul 1, 2014 at 5:47
  • $\begingroup$ Christian; What you said means the closure of $\mathrm{Int}(\mathbb N,\mathbb Z)$ in $C^0(\mathbb C,\mathbb C)$ is at most the set of continuous functions on $\mathbb C$ that takes integer values on $\mathbb N$ whose restriction is real valued on $\mathbb R$. So now the question is: are the sets equal? $\endgroup$
    – joaopa
    Commented Jul 1, 2014 at 8:34
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    $\begingroup$ Actually what he's saying is that the closure (wrto local uniform convergence on $\mathbb{C}$) of the set of the complex polynomials mapping $\mathbb{Z}$ into itself, is included in the set of real entire functions mapping $\mathbb{Z}$ into itself. $\endgroup$
    – Pietro Majer
    Commented Jul 1, 2014 at 14:19

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