> There are probably even good examples of functions that are not differentiable on any open interval but are everywhere continuous, but I can't think of any right now.

You can find loads of them once you go into the complex plane. I haven't looked into this, but I have heard that you can get functions that consist entirely of corners, so are continuous everywhere but differentiable nowhere. I think we cover complex Calculus next year, so I'll keep you posted. There's nothing in any of my books on it, unfortunately.

Matthew