Hi,


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 $n$"? 


Thanks!



An application of Baire category theorem
For a proof of the much stronger result indicated above, see Page 53 [here](http://books.google.co.il/books?id=P30Y7daiGvQC&pg=PA53&lpg=PA53&dq=%252522function+is+a+polynomial%252522+donoghue&source=bl&ots=CUDjO7Sd8m&sig=ahp6yFK0fO7kUkhgLCiFOH6OgZ8&hl=iw&ei=fSEqTbHeNsTFswby5JjhAg&sa=X&oi=book_result&ct=result&resnum=1&ved=0CAYQ6AEwAA#v=onepage&q&f=false); 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.


