I was having a discussion with some colleagues and the topic of whether or not the universe is continuous/discrete came up: we all came to the same conclusion that no matter the situation, we'll never be able to know for sure (because to prove it one way or the other would require having measurements of infinite precision, which is impossible). A colleague gave a rough/informal example of this: he claimed an observer living entirely within $\mathbb{N}$ (imagine the same picture we have for surfaces of an ant walking along the surface, but for $\mathbb{N}$ instead) could never prove his universe to be discrete.
We, habitants of $\geq 3$ dimensional space, naturally know $\mathbb{N}$ is discrete and consists entirely of totally isolated points. I thought about it for a while and all the ways we can usually prove $\mathbb{N}$ is discrete cannot be turned into "experiments" an observer living entirely inside $\mathbb{N}$ could perform (for example, an observer living in the space $\mathbb{S}^1$ could perform the experiment of walking around a finite distance in a finite amount of time and see they have returned to the same starting place, thus carrying out an intrinsic proof that $\mathbb{S}^1$ is not homeomorphic to $\mathbb{R}$).
Now, I realize this is all a bit meaningless since I for one don't see how an observer living entirely within $\mathbb{N}$ could ever leave his starting place to begin with (since all points of $\mathbb{N}$ are totally isolated). But for argument's sake, please dismiss this hurdle. To give a more formalized and tractable version of the problem, consider $\mathbb{N}$ embedded in $\mathbb{R}^3$ as $$\mathbb{N} \times \{(0,0) \} = \{ (n, 0, 0) \ \vert\ n \in \mathbb{N}\}$$
Is there any experiment a habitant of this space could perform to prove that his universe is discrete (say, not homeomorphic to $\mathbb{R}$ seen as the line $\{(t, 0, 0) \ \vert \ t \in \mathbb{R}\}$)?
Well, the world is not such and such, it's our models of the world that are such and such. But of course some models are more graceful than others so they are preferred. As an example, the 19th century concept of ether as the propagation medium for light, was discarded because it brought nothing and was complex to explain, not because it could be possible to disprove it. Similarly, there are some deterministic interpretations of quantum theory, but they are not mainstream because they bring nothing more and are more complex.
If the world - or, more exactly, our experiments results - are correctly explained by a theory with a continuous space, assuming it uses only space-continuous variables you can always make a discrete-space theory that gives the same results, with a sufficiently finely grained space. Under the assumption that our experiments have a limited precision, of course.
Reciprocally, if our experiments are correctly explained by a discrete-space theory, you can always invent a continuous-space theory and say the space is filled with a grid of particles that explain the seemingly discrete results. (Actually the Higgs Boson looks somewhat like that kind of particle - but I am not a physicist at all). Or use any math model that generates discrete phenomena despite the space being continuous (e.g. eigenvalues). Whether such a theory would be graceful enough to survive is not obvious, so perhaps in that case the discrete nature of space would be considered proven.