I'm trying to understand why monotone functions can only have jump discontinuities. I've seen formal proofs, but I don't find them intuitively convincing; my imagination can still conceive of an increasing function which is just a bunch of scattered points. To take a simple case, why can't you have a monotone function whose range only contains rational numbers, or only contains irrational numbers? Or to take a more complicated case, Conway's base 13 function satisfies the conclusion of the intermediate value theorem even though it is nowhere continuous. Is there some kind of operation which "pushes up points" on the graph of this function which would make it increasing? Any help would be greatly appreciated. Thank You in Advance.