Consider $h>0$, I would like to know what are the functions $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfy the following relation
$$f(x) = \frac{f(x+h) - f(x)}{h}. $$
NB: I do not even have the guarantee of the continuity of $f$.
The only thing that concludes was that $$f(x) = \frac{f(x + nh)}{(1+h)^n}, $$ but I do not find it useful.
Hint: note that the value $f(x)$ lets us determine all the values $f(x+k \cdot h)$ for $k \in \mathbb{Z}$ and doesn't affect any other value. Consider the equivalence relation ${\sim} \subseteq \mathbb{R} \times \mathbb{R}$
$$x \sim y \iff x-y = k \cdot h \text{ for some } k \in \mathbb{Z}.$$
Finding a function $f : \mathbb{R} \to \mathbb{R}$ satisfying the given equation is equivalent to finding $f \upharpoonright [x]_{\sim}$ for all $x \in \mathbb{R}$ separately and then combining them into one function $: \mathbb{R} \to \mathbb{R}$.