## Sufficient Condition to be a Polynomial?

Appropriate Level:  Real analysis.

The Question:  Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ is infinitely differentiable and that for each $x \in \mathbb{R}$ there is some $n \in \mathbb{N}$ for which the $n$-th derivative of $f$ at $x$ equals 0.  Must $f$ be a polynomial?