Is there a use for discontinuity? In many theorems and applications, continuity is important to get certain results. So the idea is to get continuity in your solution.
Is there a reason you want discontinuity? 

Not really - from a purely mathematical perspective, "discontinuity" is just the absence of a convenient property. If math were just about looking pretty, most mathematicians would just pretend everything was continuous.
However, discontinuous functions happen. It's like asking if there's some reason you'd want to have traffic jams - of course not, but they happen anyway, so we have to know how to deal with them. Discontinuities show up in applied math all the time; they even show up in "pure" (i.e., non-applied) situations, because there are some functions (e.g., $\sqrt{x}$) which are natural but inherently discontinuous.
