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There's a problem from calculus I remember: $$\forall x\ \exists n.\ f^{(n)}(x) = 0 \iff \exists n\ \forall x.\ f^{(n)}(x) = 0\,.$$

Function $f \in C^\infty(\mathbb{R})$, and the notation $f^{(n)}$ means differentiation.

Right side is just curious statement that $f$ is a polynomial. Of course $(\Leftarrow)$ is just trivial, however, $(\Rightarrow)$ is far from obvious. Have anybody seen this, maybe somebody knows where it comes from? What about the proof of $(\Rightarrow)$?

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    I've seen it once on the forum http://www.les-mathematiques.net/phorum/. We have to use the Baire category theorem. But I forgot the proof.2012-03-12
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    oh yeah, it's here http://mathoverflow.net/questions/51581/an-application-of-baire-category-theorem :D2012-03-12
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    See also my posting in https://groups.google.com/group/sci.math/msg/8963982857bc5f31?hl=en2012-03-12
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    The version on MO closest to this question is http://mathoverflow.net/questions/34059/if-f-is-infinitely-differentiable-then-f-coincides-with-a-polynomial2012-03-16

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Here are some reference about the problem: