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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 $\leq n-1$ ?

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What is the research question in which this problem arose? – Michael Renardy Nov 17 '12 at 17:01
up vote 1 down vote accepted

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 $n-1$.

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