Let X be an uncountable subset of ℂ and let f : ℂ → ℂ be an entire function.
Assume that for every z ∈ X, there exists an integer n ≥ 1 such that f(n) (z) = 0. Which of the following statements are necessarily true?
1
f = 0.
2
f is a constant function.
3
There exists a compact subset K of ℂ such that f-1(K) is not compact.
4
f is a polynomial.