Is reality continuous or discrete?
Reality is whatever it is… only our models of it can be considered continuous or discrete (or true, or false, or useful).
I say this like it’s obvious, but it’s a potentially controversial opinion. 🙂
People have a strong tendency to confuse the map with the territory. So a very successful theory becomes synonymous with reality itself.
But things get murky when we investigate all the details of the theory. If we are being extra cautious about what we consider “real”, then we can always wait for experimental confirmation before believing in the existence of some thing or process proposed by a theory.