Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Prove: If $f\in C^{\infty}[a,b]$, and $\forall x\in[a,b]$, $\exists n\in \mathbb{N}$, $\forall m>n$, $f^{(m)}(x)=0$, then $f$ is a polynomial.

It's a problem I got from Zorich's Mathematical Analysis Vol.II, Chap 9.5 Prob. 2(b).

share|cite|improve this question
Does $n$ depends on $x$? – Arctic Char Nov 20 '13 at 7:24
Yes, for different $x$ there is different $n$. – Andy Nov 20 '13 at 7:30
Perhaps overkill, but this can be done using the Baire Category theorem (with weaker assumptions, I'd add). – user61527 Nov 20 '13 at 7:33
@T.Bongers: From the title might be it is not an overkill? ^^ – Arctic Char Nov 20 '13 at 7:34
Thanks a lot! This is exactly the solution I am looking for! – Andy Nov 20 '13 at 7:42

This is a community wiki answer designed to help eliminate this question from the unanswered queue

The OP was totally satisfied by this solution to the problem on mathoverflow.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.