I often heard about this division but always in a non-formal manner. What constitutes it? Is it a limit operation? Or a fundamental distinction between countable and uncountable sets? And what math subjects can be accurately attributed to one or another type?


I really find this division very weird.



What constitutes the division between discrete and non-discrete math? Are there any math subject where it's being blurred?
There is no real division. It is for convenience. An example in which you can test out ideas on this is with the theory of finite topological spaces. (NB every thing is finite so the answer to your last part is clear in this context.) Finite topological spaces when discrete are finite sets (full stop nothing more can be said) when $T\_0$ then they begin to resemble preordered sets. There is a beautiful homotopy theory of finite topological spaces and it is (almost) as rich as ordinary homotopy theory. Check it out and the division discrete non-discrete is almost quantifiable here, yet it is also very close to not existing as partially ordered sets... that is clear that is part of discrete mathematics, whilst think of them as topological spaces and they are not discrete at all.


