The question is what exactly is continuous. The quantum fields themselves? And we have no evidence the the quantum fields are discrete quantum states themselves, but the particles they produce appear otherwise.
It's always possible that continuous maths are a side effect of this field. Of course, the causality could be revered.
if you think of universe as minecraft lattice (which is arguably best way to think of it and explain all of physics currently in existence in intuitive level) then yea, it can be like eg a spin lattice discrete at some plank length from which field perturbations as particles emerge..look up spin lattice from https://arxiv.org/abs/hep-th/0507118 so basically bringing back idea of quantum ether cuz explaining empty space with waves without some underlying structure doesn't really make sense anyway since all other waves have mediums in nature.
It is an interesting discussion you bring up but will take much more than a comment to talk about. There is a divide between mathematics and physics regarding numbers. Real numbers have supposedly infinite precision, and arbitrary size, which are not true in our physical universe. Indeed infinities have given physicists quite some headaches. It's a bit of a dark art as to where you can and cannot just use infinity willy nilly and many have expressed distaste to it. Also at some point you have to talk about computability, like are these infinite precision and magnitudes even real if nothing in the universe even gets close, what is omega to the omega, statements dream up by the utterly deranged?
I remember there being a talk or some article about this but can't find it anymore, but it is an interesting thing.
Only infinite magnitude causes problems in physics, infinite precision doesn't. Real numbers have infinite number of digits only in digital model of numbers, so their infinite size is a modelling artifact. e=1 in base e - one digit.
It's always possible that continuous maths are a side effect of this field. Of course, the causality could be revered.