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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 We have to use the Baire category theorem. But I forgot the proof. – Đức Anh Mar 12 '12 at 21:26
oh yeah, it's here… :D – Đức Anh Mar 12 '12 at 21:28
@canh Thank you! – dtldarek Mar 12 '12 at 21:43
See also my posting in – Robert Israel Mar 12 '12 at 22:27
The version on MO closest to this question is… – Will Jagy Mar 16 '12 at 23:05

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