$\newcommand\Z{\mathbb Z}\newcommand\R{\mathbb R}$Let 
$$f(x):=\sum_{n\in\Z}2^{-n}K\Big(\frac{x-2^{-n}}{2^{-n-2}}\Big)
=\sum_{n\in\Z}2^{-n}K\Big(\frac{x-2^{-n}}{2^{-n-2}}\Big)1(x\in2^{-n}[3/4,5/4]),$$
where $K$ is a smooth function such that $K(0)=1$ and $K(x)=0$ if $|x|\ge1$. Note that the intervals $2^{-n}[3/4,5/4]$ are disjoint for distinct integers $n$. 

So, $f$ is smooth on $\R\setminus\{0\}$, $f(0)=0$, and $f(2x)=2f(x)$ for all real $x$. So, for $n=2$,
$$\lim_{h\to 0}\frac{\sum_{k=0}^n f(x+k h)\binom nk (-1)^{n-k}}{h^n}$$ 
exists for all $x\in\R$, but $f$ is not even differentiable at $0$, because $f(2^{-k})=2^{-k}$ and $f(\frac34\,2^{-k})=0$ for all natural $k$.