About the so-called "discontinuous truth values", I think the culprit is not that the truth value is discontinuous (it doesn't make sense to say a point-value is continuous or not), but rather that we have a binary predicate, "less-than", which is a discontinuous truth functional mapping.

The statement "less-than(tv, 0.5)" seems to be a binary statement. If we make that predicate fuzzy, it becomes "approximately less than 0.5", which we can visualize as a sigmoidal curve, and this curve intersects with the slope=1 line at 0.5. Thus, the truth value of the fuzzy version of that statement is 0.5, ie, indeterminate.

All in all, this problem seems to stem from the fact that we've introduced the binary predicate "less-than".