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Does somebody know a proof (or a reference) for the following statement:

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be an infinitely differentiable function. Suppose that for all $x$, $f^n(x)=0$ for almost all $n$ (i.e. for all but finitely many $n$), $f^n$ being the $n$-th derivative of $f$. Then $f$ is a polynomial function.

From what I remember, it is a result of Sunyer i Balaguer, and involves the use of Baire category theorem, but I cannot find any reference on the web.

Also, is the theorem still true if one replaces "almost all $n$" by "infinitely many $n$"?


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Actually a much stronger statement is true:… – Qiaochu Yuan Jan 9 '11 at 20:27
up vote 3 down vote accepted

For a proof of the much stronger result indicated above, see Page 53 here; the theorem states the following:

Let $f(x)$ be $C^\infty$ on $(c,d)$ such that for every point $x$ in the interval there exists an integer $N_x$ for which $f^{(N_x)}(x)=0$; then $f(x)$ is a polynomial.

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Great thanks! Indeed the actual theorem is much stronger than what I thought was true. – Laurent Bienvenu Jan 10 '11 at 8:01

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