Properties of entire functions of which the $n$-derivatives have uncountably many zeros

Question

Let $f$ be an entire function and $$A = \{z \in \mathbb C: f^{(n)} (z) = 0 \text{ for some positive integer } n\}.$$ Then what can we say about the function $f$ if $A$ is uncountable?

Answer 1

The zeroes of an analytic function are isolated; and any uncountable subset of the plane has an accumulation point. So, there is some $n $ with $f^{(n)}=0$, and thus $f $ is a polynomial of degree $n-1$ or less.