Let $f:R\rightarrow R$. If there exists the finite limit $$\lim_{(x,y) \rightarrow a \atop x\neq y} \frac{f((y)-f(x)}{y-x}$$ then obviously there is a finite derivative $f'(a)$ and is equal this limit.

What about similar problem for higher order divided differences?

May is it true that existence of finite $$\lim_{(x_0,...,x_n)\rightarrow (a,...,a) \atop x_i \neq a} [x_0,...,x_n;f]$$ implies existence of $f^{(n)}(a)$?

If not is there connetion between high order divided differences and derivatives?