heine borel and anything relying on compactness fails.
weirdest/coolest result is IVT fails too, which is bizarre because on R continuity is strictly stronger than IVP, but in dense-but-not-complete ...
Funny, I went to bed last night wondering about IVT, as it was about the only result on continuous functions I could come up with. Will have to ponder it more to see why that may be though.
Heine Borel isn't a statement about _functions_ though, unless I'm mistaken, so hadn't considered it in this context.
Keep the math posts coming! Oh, btw, ever looked into the surreal numbers? Literally started reading an article about continuity on them the other day, and how the concept fails in surprising ways, but also works in another. Haven't finished it yet, so don't even know exactly what the conclusion is, but seemed worth mentioning.
Heine Borel isn't a statement about _functions_ though, unless I'm mistaken, so hadn't considered it in this context.
Keep the math posts coming! Oh, btw, ever looked into the surreal numbers? Literally started reading an article about continuity on them the other day, and how the concept fails in surprising ways, but also works in another. Haven't finished it yet, so don't even know exactly what the conclusion is, but seemed worth mentioning.