# An Entire function with a condition. [closed]

Let $f$ be an entire function. Suppose, for each $a\in \mathbb{R}$, there exists at least one coefficient $c_{n}$ in $f(z)=\sum c_{n}(z-a)^{n},$ which is zero. Then prove that there exist $k\geq 0$ such that $f^{(n)}(0)= 0$ for all $n\geq k.$ Where $f^{(n)}(0)$ means its n th derivative at zero.

## closed as off-topic by Daniel W. Farlow, anomaly, Alice Ryhl, Christopher, user98602 May 12 '15 at 18:47

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Daniel W. Farlow, anomaly, Alice Ryhl, Christopher, Community
If this question can be reworded to fit the rules in the help center, please edit the question.

• The given condition just says that at each reals either function value is zero or its derivative is zero at that point. Then how to prove that after some stage function derivative is zero at zero? – neelkanth May 12 '15 at 16:07
• Please do not simply dump your homework problems here. Instead, please indicate what you're stuck on, what you've tried so far, and the context of the question. – anomaly May 12 '15 at 16:34

Hint: for $n=0,1,2,\cdots,$  let $E_n= \{a \in \mathbb {R}: f^{(n)}(a) = 0\}.$ Argue that one of the $E_n$'s is uncountable and proceed. (There's a proof using Baire here, but we don't need to use that.)