Let $U\subset\mathbb R$ be an open set. Let $n\in\mathbb N$ and suppose that $f\in\mathcal C^n(U)$, i.e. that $f$ is $n$-times continuously differentiable on $U$. The $n$-th derivative of $f$, denoted by $f^{(n)}$, then satisfies, for all $x\in U$, $$f^{(n)}(x)=\lim_{h\to 0}\frac{\sum_{k=0}^n f(x+ k h) \binom nk (-1)^{n-k}}{h^n}.$$

I proved this fact here and the result is closely related to the Grünwald–Letnikov derivative.

**My question.** Suppose that $f: U \to\mathbb R$ is *any* function such that $$\lim_{h\to 0}\frac{\sum_{k=0}^n f(x+ k h) \binom nk (-1)^{n-k}}{h^n}$$ exists for all $x\in U$. Does it follow that $f$ is $n$ times differentiable? If yes, is there an easy proof or a reference that proves this?