I meant that setting the limit to no preference for a given C doesn't equate to a globally continuous function. But that when you adjust your preferences function to approximate the discontinuous function by a continuous one, the result will contain (at least one) no preference point between any two A < B.

Now perhaps there is a result which says that if you take the limit as you set all discontinuous C to no preference, that the resulting function is complete, consistent, transitive, and continuous, but I wouldn't take that to be automatic.

Consider, for example, a step discontinuity, where an entire swatch of pA + (1-p)B are stuck on the same set of < and = mappings and then there is a sharp jump to a very large set of < and = mappings at a critical p'. If you map the ordinals to the real line, this is analogous to a y-coordinate jump. To remove this discontinuity you would need to do more than split the preferences at p' around no preference, because all this does is add a single point to the mix. To fully resolve it, you need to add an entire continuous curve, which means a process of selecting new A, B, and C, and showing that the transfinite limit always converges to a valid result.