The continuity hypothesis really is an unimportant "technical assumption." The only kind of thing it rules out are lexicographical preferences, like if you maximize X, but use Y as a tie-breaker.

Specifically, it follows from independence that if A<B<C, then there is some P so that pA+(1-p)C is better than B for p<P and worse than B for p>P; the only thing the continuity axiom requires is that at P there is no preference between B and the mixture; there is no tie-breaker. (Without the continuity axiom, it may well be that P is 0 or 1.)

This is still true if you only have preferences involving p a rational number: the above is a Dedekind cut. If you restrict p to some smaller set that isn't dense, it's probably bad, but then I'd say you aren't taking probability seriously.