Suppose we ascertained that space and time are discrete and the units are Planck's. Would that affect calculus? I know that integration does not require a continuum, but about differentiation I read contrasting views, w.r.t. physical functions.
In order to be differentiable a function must be continuous, does that imply that dx must be a continuum?
Can someone say a final word?
"Would that affect calculus?" Of course not, and calculus in its present form would even remain a perfect tool for describing the macroscopic world. Note that, e.g,. we already know that matter comes in the form of "little balls", but nevertheless we treat it as a homogeneous "glue" when we do elasticity or hydrodynamics.
On the other hand it might be the case that some sort of new mathematics would be needed to describe and understand better time and space in its ultimate fine grained structure.
A discrete universe doesn't mean we wouldn't describe it continuously macroscopically, but we could do calculus discretely if we wanted - generally speaking it just means leaving in higher orders of the increments instead of discarding them. 'Some theorists are more willing to speculate than others. But even the boldest acknowledge the "Planck scales" as an ultimate barrier. We cannot measure distances smaller than the Planck length [about 10^19 times smaller than a proton]. We cannot distinguish two events (or even decide which came first) when the time interval between them is less than the Planck time (about 10^-43 seconds).' Just Six Numbers, Martin Rees; quoted - p317, The Continuous and the Infinitesimal, J L Bell.
Calculus is not a description of the physical world, but a description of an idealization of the physical world. In that idealization we regard space and time as a continuous space. Whether space and time are actually continuous -- or, for that matter, whether the universe is finite or infinite, bounded or unbounded -- is essentially irrelevant to calculus.
On the other hand, one might ask a slightly different question: If the universe is discrete at the Planck length, would Calculus still be useful for describing it, and if not, would there be a more suitable alternative? To this question, the answer is: Certainly Calculus would still be useful for describing the universe at scales much larger than the Planck length, because at those scales the discrete nature of space and time is essentially imperceptible. As one approaches those very small scales, the error in modeling space and time as a continuum would begin to mount, and predictions made by the idealization would stop matching up with reality.
But there is no need to ask "What if?", because in fact physicists have been dealing with this for decades. Much of quantum field theory is done using a model of the universe in which spacetime is discrete at very small scales; this is known as an ultraviolet cutoff. The name comes from the fact that small lengths correspond to high energies and frequencies; similarly, a model of the universe in which spacetime has some finite size (maximal lengths) is called an infrared cutoff, because large lengths correspond to low energies and frequencies. There is a short article about both cutoffs on Wikipedia and there are a number of questions about them on the Physics Stack Exchange site.
In a discrete world, one would still have the question of finding $$ \lim_{n\to\infty} \frac{\pi(n)}{ n/\log n }, $$ (where $\pi$ is the prime-counting function) and there you have calculus.
(And other results in analytic number theory too.)
