This is both one of the most important questions to ask in physics and also one of the most ridiculous.
Many of the arguments go like this: we can't do the math if it's not continuous therefore it's continuous, or I don't get how time can be discrete so it must be continuous, or "history and tradition says... so please don't upset things."
If a system consists of discrete samples, then that is isomorphic to another system using continuous functions under the condition of finite information content. In that case, they are just different basis sets, one consisting of delta functions and another consisting of a superposition of any reasonable orthogonal basis set. Being made of discrete points in space or time is the same as saying the system is band-limited in Fourier space and can't have infinite energy, (or information content established below). It's a matter of choice then whether you want to consider the system as being made of continuous functions or an array of numbers.
That's why, for example electrons stuck in a potential well have discrete energy levels representable as a finite set of quantum numbers, because of finite boundary conditions and the enforcement of integer modes of wave-function continuity around a sphere, but given sufficient extra energy they're off somewhere else entirely.
This map from discrete to continuous is standard signal processing math of sampled systems which is used everyday whenever you listen to music etc. It goes from numbers in a computer to air vibrations, and mathematically can be done without any loss of information (no compression need be used in the music example.)
We might use a spatial or temporal basis of samples and a Fourier basis, or wavelet basis, or a mixture of Gaussians, etc. Every set of numbers that purports to describe a physical system makes no sense without an accompanying definition of its basis. And for each such basis there is a linear method to provide the lossless reconstruction of its representation as a continuous function, whether real or complex.
So it's always possible to map a continuous description of any finite bounded system onto a discrete description of the same system by some unitary transform which places zero everywhere except at a finite set of points. So the original question amounts to "how long is a piece of string?"
The real question is: Does the universe have a finite information content? Is the universe bounded? But more importantly: is the universe locally bounded with finite local resources, or a finite rate of accumulating local resources to meet demand?
The relevant thing to ask next is whether the universe has a finite maximum information density. That is: Can there be an infinity of points, arbitrarily close together each containing an infinite resolution of numerical values and in communication with their neighbors, with infinite bandwidth of their ability to transfer precisely those values over space and time without error from place to place in infinitesimal time, OR NOT.
It's worth bearing in mind that if you assume a continuum to time and space without any other consideration, you're loading in the assumption of infinite local resources and infinite information content, and to me Occam's razor say's no no no.
But evidence from physics also does not support this.
Firstly, there's an upper limit to energy density, before space collapses to a black hole singularity.
There's the related holographic entropy bound, which limits the maximum amount of bits that can be represented across any surface area and implicitly the minimum space resolution.
There's the Landauer limit which is established by experiment to say that it takes around 2.805 zJ to flip the information state of a physics system by one bit. It varies with temperature I believe, which is a little like inverse signal to noise ratio. So you're going to have to use more energy to flip more bits in any system. Each one comes with a finite cost. So it seems that energy cost prevents the packing of infinitely precise numbers arbitrarily densely if any of them will ever change.
And also there's the fact that it takes a finite non zero amount of time for any quantum system to transition from one orthogonal state to another orthogonal state and that provides a temporal bound on the upper limit of the rate of computation of any physical system. The processing rate cannot be higher than 6 × 10^33 operations per second per joule of energy according to the Margolus–Levitin theorem. And we can't put in as much energy as we'd like to make things go faster because of the previous constraints.
So to me that implies that to speak of a physical system computing things faster than a bounded rate is not physically meaningful, and so the universe can't simulate itself faster than the rate in which it can move between orthogonal states which represent information that is at some point isolated and distinct and discriminable as a set of observables, as opposed to being always entangled.
There's also evidence from the scattering matrix and its growing association with permutations: cayley graphs on small groups, the permutahedron, and finite rational sequences, that particles themselves are in some space similar to low dimensional polytopes, with discrete geometric representations.
And finally there's a question of scale relativity. We have special relativity but we don't know if there's a preferred scale and we observe everything from one specific preferred vantage point. There are publications by serious physicists advancing the argument that scale invariance is part of relativity puzzle (e.g. Laurent Nottale's book on scale relativity) and that means that the "sampling grid" if any, could appear to vary from place to place, or between observers under different conditions, because we don't have any absolute yardstick for size scale. I jokingly like to say: do you realize you're actually only 1/10th the size that you were when you were born? If everything scales freely over time slowly how would we ever know? Only extremely rapid differential changes of scale in a fully relativistic universe would create noticeable effects for humans. There should be a Higgs boson equivalent for local relativistic scale changes.
There are more arguments that come to mind but this a discrete and very bounded sample of them.