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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?

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We can name a lot of reasons why one should prefer differentials to finite differences, but I guess one of the most physical ones is the relativity!

We assume that there is a finite speed limit for information transfer, that is the speed of light, hence any physical quantity should be local, in the sense that its constituents do not require instantaneous interaction.

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 $x+h$!

Oh, by the way, this is a problem if we use difference instead of differential while assuming that spacetime is continuous. If we also assume that spacetime is discretized, then we need to modify Special Relativity (SR): As it stands, SR states that different observers measure lengths differently, hence there cannot be a naive universal minimum length. There are theories which break SR (for example Doubly Special Relativity), but the vast mainstream physics, and almost all of its foundations, require differentiation rather than differences.

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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.

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