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?