{f(x) belongs to R[x] | f(n) belongs to Z for all n belongs to Z}uncountable or countable ? (TIFR GS 2022) [closed]
The set of polynomials with real (or complex) coefficients which map integers to integers is larger than $\mathbb{Z}[X]$, but not much larger. It is the set of $\mathbb{Z}$-linear combinations of the rational polynomials $\binom{X}{m}$ for $m=0,1,2,\ldots$. Here the “binomial polynomial” $\binom{X}{m}$ is simply
$$\binom{X}{m} = \frac{X(X-1)\cdots(X-m+1)}{m!}$$
Since every such polynomial is a finite linear combination of these basic polynomials, and the coefficients belong to a countable set, the set of such polynomials is itself countable.