In Rudin's Principles of Mathematical Analysis, he first proves that the set of points at which a monotonic function is discontinuous is at most countable (Theorem 4.30). Immediately following this proof, he remarks (Remark 4.31) "It should be noted that the discontinuities of a monotonic function need not be isolated. In fact, given any countable subset E of (a,b), which may even be dense, we can construct a function f, monotonic on (a,b), discontinuous at every point of E, and at no other point of (a,b)."
I don't quite follow the logical flow here. What is the meaning of discontinuities not needing to be isolated? Does it mean that the set of points at which the function is discontinuous may have points that are limits points of the set?
Is there a specific reason he is clarifying this via a remark? Does Theorem 4.30 mislead people to think that the discontinuities would be isolated? If so, why?
Finally, what is the significance of the phrase, "which may even be dense" in the remark? Does being dense have anything to do with having points that aren't isolated points?