Real analysis shows that the real line is complete. All the gaps between the rationals are filled. It is perfectly continuous. Of course, all of this is based off of certain axioms that make it work mathematically. Are those axioms now sound? According to quantum, the plank length is the smallest unit of space allowable in existence. So if the idea of continuity is flawed in reality, then the current state of much of mathematics is also on shaky grounds.