From: israel@math.ubc.ca (Robert Israel)
Newsgroups: sci.math
Subject: Re: IT IS A POLYNOMIAL
Date: 5 May 1998 20:27:44 GMT

Alexander Abian wrote:
> Can you give  as clear and as simple as possible proof of the following
> in the Real Analysis?
> 
> Let  f  be an infinitely differentiable function  on   [0, 1]  such
> that at every  x  in [0, 1]  an n-th order derivative of  f  is zero,
> where  n  depends on  x.  Prove that  f  is a polynomial.

It brings back memories: this problem (the "polynomial problem") was going 
around U. of Chicago ca. 1970.  

Suppose f is not a polynomial.  Let 
C = {x: there is no neighbourhood of x on which f is a polynomial}.
A_n = {x: f^(n)(x) = 0}.

Clearly C is closed and nonempty, and A_n closed with union_n A_n = [0,1].
Applying the Baire Category Theorem to C, A_n intersect C has nonempty interior
in C for some n, i.e. there exist x0 in C and delta > 0 such that
C intersect (x0 - delta, x0 + delta) is contained in A_n.  

Suppose f^(k)(x0) <> 0 for some k > n.  By Taylor's Theorem, we get f^(n)(x) <> 0
for 0 < |x - x0| < eta (for some eta > 0).  Taking eta < delta, this implies
C intersect (x0 - eta, x0 + eta) = { x0 }, but then f is a polynomial on
(x0 - eta, x0) and on (x0, x0 + eta) and hence on (x0 - eta, x0 + eta),
contradicting x0 in C.  So we must conclude that f^(k)(x0) = 0 for all k >= n.
The same is true for all points in C intersect (x0 - delta, x0 + delta).

Now suppose c in (x0 - delta, x0 + delta) \ C.  Then f is a polynomial on some
neighbourhood of c.  Let [a,b] be the maximal interval containing c on which
f is a polynomial.  Then a or b (say b) is in 
(x0 - delta, x0 + delta) intersect C, so that f^(k)(b) = 0 for all k >= n.
Now if f has degree p on [a,b], f^(p) <> 0 on [a,b] so p < n, and then
f^(k)(c) = 0 for all k >= n.  

We conclude that f^(k)(x) = 0 for all k >= n and all x in (x0 - delta, x0 + delta).  But then f is a polynomial on (x0 - delta, x0 + delta), contradiction.

Robert Israel                            israel@math.ubc.ca
Department of Mathematics             (604) 822-3629
University of British Columbia            fax 822-6074
Vancouver, BC, Canada V6T 1Z2