Neither proven nor disproven in any capacity.
Also I have no clue how anyone could assert that the universe being "continuous" (whatever that means) could be casually claimed by anyone to have any link to our universe being "a simulation" (whatever that means) or not.
So to simulate continuum you need a transfinite VM. It will be more efficient to just run the universe bare metal.
But as far as I understand that is just the result of constraints imposed by the atom/molecule "box". Like how a guitar string can only vibrate at certain frequencies because an integer multiple of wavelengths of the standing wave must fit on the string.
Outside of such systems, energy levels are not quantized. For example, photons from distant galaxies appear to be redshifted on a continuous spectrum.