Certain properties of particles, like velocity, are continuous. (Note, if velocity were not continuous, acceleration would be problematic.)
Other properties, like frequency, are continuous. The blackbody radiation, for example, spans a continuum of frequencies.
There is no velocity (less that c) that is prohibited. There is no frequency that is prohibited. There is no requirement for either of these to be restricted to some integer series.
So reality can't be discrete.
As for reality as a simulation, that seems more like a question for Sister Morphine, the mind of Buddha, or Billy T's parietal lobe. I don't make any such presumption. Suppose reality just is what it is: mostly a continuum, plus a quantum world, plus a virtual world (not meaning a simulation), plus whatever creates the arrow of time, plus whatever bends time (and space and mass), plus whatever creates the continuum, quantum reality, virtual reality, relativity, entanglement &etc., etc., etc. In particular, suppose reality really includes a Big Bang that creates spacetime and therefore always was.
As hard as it is to get my arms around all of that, there is at least "best evidence" to draw conclusions from, to trace the nature of reality. I just can't get my arms around the idea of simulation any more than I can get them around the idea of a Supreme Commander of the Universe who, no doubt, is at the joystick, fumbling for another quarter to stick into the slot before time's up. I'm all for legalization of pot, I just don't think it's the breakfast of realists.
Thanks for the explanation, Aqueous.