Question :
Let $f$ be an entire function . Suppose for each $a \in \mathbb R$, there exist atleast one coefficient $c_n$ in $f(z) = \sum_{n=0}^{\infty}c_n (z-a)^n$, which is zero . Then
$f^{(n)}(0) = 0$ for infinitely many $ n \geq 0$
$f^{(2n)} (0) = 0$ for every $ n \geq 0$.
$f^{(2n+1)} (0) = 0$ for every $ n \geq 0$.
there exist a $ k \geq 0$ such that $f^{(n)} (0) = 0$ for all $ n \geq k$.
we know that the zeroes of a non constant analytic function is a discete set which is contable . So zeroes of each derivative set is countable . So its union is countable . If $f$ is not a polynomial, then we get a contradiction according to the statement of the question.
Thus $f$ is a polynomial. So (1) option and (4) option are true.
For (2) , $f(z) = z^2$, For (3), $f(z) = z^3$ are counter example.
I would be thankful, if someone check my Solution.
Thank you
