What if space-time *isn't* in continuum form?
We definitely seem to perceive it that way, and so far it's given us pretty good results. There's mutterings of actual granularity, but i find them distasteful (look up stuff on the Planck length if you're really, really, seriously, going-to-devote-hours interested). Smoothness feels Right, but, yah, we probably live in a very finely granular Universe. Which is ok enough, i suppose.
But that's not the kind of non-continuum you were talking about, i suspect.
I can't think of any good way to test the theory of universal discontinuity while one is part of the system, since things would be constantly leaping around,but only to internally-consistent points. Living in a system that randomly jumped to inconsistent points would be very nice: it'd give me something to blame my piss-poor short-term memory on! =)