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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.

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    *Weierstrass*, with two *e*'s and no *u*.2012-03-12
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    I don't understand what you're suggesting. You say you want an example of a uniformly continuous but nowhere differentiable function $f: \mathbb{R} \rightarrow \mathbb{R}$, but the example you give is of a function with domain $[a,b]$. How can this be what you want?2012-03-12
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    Also, what is the "Bolzano-Weirstrauss function"?2012-03-12
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    @Pete There's a lack of clarity here on my part,my bad. I meant a real valued function defined on any subset of the real line.Secondly,obviously I meant the Weierstrass function. Bolzano proved such a function existed,but he never gave an explicit procedure for constructing one.Again,my fault entirely,going to edit the post so it's clearer.2012-03-12
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    e.g., i.e. - whatever :)2012-03-12
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    @The Chaz Uh,no,if it was "whatever" I would have just ignored Pete's post. Clearly I thought he had real points,don't you think?2012-03-12
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    @Mathemagician1234: I am pretty sure The Chaz is not accusing you of ignoring anyone. He's just pointing out that you wrote "i.e." where you probably should have written "e.g." (@The Chaz: in the realm of pedantic -- but of course correct -- comments on people's writing, this is a pretty "advanced" one. If you have the energy to spare and want to help, I suggest you get on the OP to put spaces after his commas and doublecheck "ie" versus "ei" in word spellings.)2012-03-13
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    What @Pete said! (I'll do my best...)2012-03-13

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