What constitutes the division between discrete and non-discrete math? Are there any math subject where it's being blurred? 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.
 A: 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.
A: Our new Professor, Stephen Gilmore, is about to give an inaugural lecture entitled something like "Is Informatics an indiscrete science?", the point being, I think, that informatics is normally thought of as a fully discrete subject (being about finite or at worst countable structures), but that once you want to talk about time or power you need non-discrete methods such as differential equations. Yes, my immediate reaction is that it's about countable vs uncountable sets. 
However, your question presupposes that the whole of maths splits into discrete and non-discrete, and I think that doesn't really reflect how the terms (term, really: discrete maths is a thing, but I'm not convinced non-discrete maths is a thing) are (is) used. Is finite group theory discrete maths? It's not what people usually have in mind, is it?
