Alright,every kid who's taken and passed undergraduate real analysis knows there are functions defined on subsets of the real line that are continuous everywhere and not differentiable everywhere (i.e.the Weierstrass function and its many descendants and bretheren). But I was going through my old analysis texts and was wondering: Is there an example of a uniformly continuous everywhere function whose domain is a well-defined nonempty subset of the real line that has no derivative anywhere?
My first response would be yes: Take the restriction of the Weierstrass function to any closed and bounded subset of the real line [a,b]. Them since this function is continuous everywhere and defined on a compact subset of the real line,then this restriction is uniformly continuous on [a,b]. Since the derivative doesn't exist anywhere on the domain of the original Weierstrass function, it doesn't exist anywhere on [a,b] either. Granted,it's not a genius construction, but as far as I can see, there are no logical errors here.
Are there? More importantly,if this example is correct, can anyone give a more creative example, one that isn't obvious? If so, I'd love to see them posted here.