This is related to Do discontinuous preferences imply no continuous utility function?
I think the title of the above-linked question is phrased in such a way that obscures a subtly different but more interesting question which the OP also hinted at in the body. I'd like to ask that explicitly here.
Does there exist a rational but discontinuous preference relation that is representable by a (potentially discontinuous) utility function?
In other words, if $\succsim$ satisfies completeness and transitivity but violates continuity, can we still find a utility function to represent it?
From known results, the answer does not seem obvious.
- We know that continuous utility representation exists if and only if preference is complete, transitive, and continuous. But this doesn't tell us what happens when preference is not continuous.
- We know that utility representation does not exist for some discontinuous preference (e.g. the lexicographic preference). But can this conclusion be generalized?
Finally, I want to note that the requirement for $\succsim$ to violate continuity means we're ruling out finite (and countable?) domains.