# Why do we use differential equations in physics instead of $h$-difference ones?

Since we don't know whether space and time are discrete or continuous wouldn't it be a better idea to use $h$-difference equations where the derivative is $$f'(x) =\frac{f(x+h)-f(x)}{h},$$ since they are more general and by sending $h$ to 0, we would have the usual differential equations. So why do we prefer differential equations instead?

The way we define, and you define above, the differences mean that they are dependent on two positions which are apart a finite distance. Hence the value of $f'(x)$ require an instantaneous interaction between the points at $x$ and $x+h$!
We use $\frac{\delta f}{\delta x}$ as "change in" or differences quite a lot. Often approximating it to a differential. The main source of discrepency between these is that the differential is based on the instantaineous gradient at x, but actually there can be some variance in gradiant if $\delta x$ is not infinitesimal. (I guess that's what you mean when you say that space-time is continuous). In other words the short answer is yes - it can be more accurate - if $\delta f$ is known e.g. measured.