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 $\leq n-1$ ?



Functions  whose divided difference is uniformly convergent to $0$ 
Yes. If the first divided difference has this property then the function is constant.
As the $n$-th divided difference is the first divided difference of the $n-1$-st divided
difference, we conclude that the $n-1$-st divided difference is constant. So your function
is polynomial of degree at most $n-1$.


