Let a function $f:(a,b) \rightarrow \mathbb R$ be continuous and such that
for each $\varepsilon >0$ there exists a $\delta >0$ such that for $x \in (a,b)$, $|h|<\delta$ such that $x+nh \in (a,b)$ :

$$| \frac{\Delta_h^n f(x)}{h^n}|:=|\frac{\sum_{i=0}^n (-1)^{n-i} \frac{n!}{i!(n-i)!} f(x+ih) }{h^n}| <\varepsilon .$$

Is it then $f$ a polynomial of degree $\leq n-1$ ?