Let $E\to M$ be a smooth vector bundle over a smooth manifold $M$. Let $D$ be a differential operator defined on the space $\Gamma(E)$ of smooth sections of $E$. We fix a section $s\in \Gamma(E)$. Assume that for every $x\in M$ there exist a natural number $n$ such that $D^n(s)(x)=0$. >Does this imply that $D^k(s)=0$, for some $k\in \mathbb{N}$? >**If the answer is "No", what about if we assume that $D$ is an elliptic operator and $M$ is a compact manifold?** The motivation for this question is the following fact which I learned from the book of "R.P.Boas:A primer of real functions". https://www.jstor.org/stable/10.4169/j.ctt5hh8x5 **Fact:** Assume that $f:\mathbb{R} \to \mathbb{R}$ is a smooth function with the property that for every $x\in \mathbb{R}$ there exist a natural number $n$ with $f^{(n)}(x)=0$.Then $f$ is a polynomial.