Every mathematical object (ok, this is false but that's not the point here) can be constructed in ZFC (the standard axiomatic framework for set theory) so you can construct the real numbers in terms of sets (if you want more precise informations on this construction look up Dedekind cuts).

However this is irrelevant to, say, analysis, you could define the real numbers as the unique (up to isomorphism) complete, ordered, archimedean field and do analysis just as well, so I'd say that you are right in some sense and some formulation, but it's a bit of a stretch to consider analysis as starting from discrete maths.

I also don't see how set theory fits into discrete maths, apart from the basics it seems pretty far from the common structures studied in discrete maths.