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 Bore...
Did a bit more thinking about your post. Good stuff. Had to think a while to come up with an example violating IVT, but a simple one would be function defined as 0 on rationals below root(2) and 1 above. I wonder if all examples look like this or if there are more complex ones I'm not thinking about.
Made me revisit some stuff on compactness, and I realized that the proof hinges (unsurprisingly) on LUB. Guess I shouldn't be surprised.
Led me to finding this tidbit from a book I like, so figured I'd share it with you.
Made me revisit some stuff on compactness, and I realized that the proof hinges (unsurprisingly) on LUB. Guess I shouldn't be surprised.
Led me to finding this tidbit from a book I like, so figured I'd share it with you.