As a takeaway of this post we have the following result.
P. Let $f:[0,1]\to\mathbb{R}$ be infinitely differentiable such that for all $x\in[0,1]$ the sequence $\{f^{(n)}(x)\}$ is eventually $0$. Then $f$ is a polynomial.
Now take $f:[0,1]\to\mathbb{R}$ be infinitely differentiable and be $S$ the set of $x$ such that the sequence $\{f^{(n)}(x)\}$ is not eventually $0$.
I got interested in answering this question: which are valid $S$? How small can $S$ be?
Property. $S$ has no isolated points.
This is a consequence of P.
Property. Let $N=\{x\in[0,1]:f\text{ is not a polynomial locally at }x\}$. Then $N$ is a perfect set (possibly empty).
Observe that $S\subset N$.
I realised that $S$ could have zero measure. In this comment I explained the construction of an infinitely differentiable function whose derivatives eventually vanish except in the Cantor set. (I.e. for which $N=\text{ Cantor set }$).
- Does anyone know of any research material pointed in this direction?
- Has the function described above appeared in the literature before?
Thank you in advance.
