The underlying problem is that infinity doesn't exist. It's a convenient illusion to make special cases go away. It's possible to have entirely constructive mathematics. In a true constructive model, everything can be constructed in a finite number of steps. There are only integers, no reals.
Well sure, there is a constructive subset of the topic we call "mathematics", just like there is a subset that admits only numbers less than or equal to 5. But if you want the whole topic of mathematics, it's going to include nonconstructive theorems like the Banach-Tarski paradox. One could take a philosophical view for or against the idea that the non-measurable set in the paradox platonically "exists", but either way, those theorems are a legitimate part of mathematics. At best you can say the theorems are about mythological entities rather than "real" ones.
Constructive mathematics can handle rational numbers.
Rational numbers are continuous, in the sense of being infinitely sub dividable.
That is, between N/M and (N+1)/M lies (2N+1) / (2M).
It's the definition of "exists" that is contested. To me, the infinity of anything is a generator (a program that continues to print digits). Does some infinite number exists "now"? Not really (it's being printed). But is the number being printed larger than any given integer? Yes. So, does infinity exists? ¯\_(ツ)_/¯
"The Emperor's New Mind" is a great book on this and related topics.
