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Are Continuous Functions Always Differentiable?
Is there a continous function (continous in every one of its points) which is not differentiable in any of its points?
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4Yes. The standard example is the Weierstrass function. – froggie Nov 22 '12 at 17:42
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1Or a Wiener process - note the "fractal" nature of the image. – kahen Nov 22 '12 at 17:44
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1Yes, most (in the sense of category) continuous functions are nowhere differentiable. – André Nicolas Nov 22 '12 at 17:49
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@AndréNicolas Could you elaborate? – Qqbt Nov 22 '12 at 17:54
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1this might help – Norbert Nov 22 '12 at 17:56
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1@Andris: Look at the continuous functions from $[0,1]$ to the reals, under the sup norm. Then the set of such functions that are continuous somewhere is meager. Standard result, I think it is in Oxtoby, but it should not be hard to track down a proof on the Web. – André Nicolas Nov 22 '12 at 18:10
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@André: your second "continuous" should be a "differentiable". – TonyK Nov 22 '12 at 18:31
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@TonyK: Thanks, yes, differentiable. – André Nicolas Nov 22 '12 at 18:42
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Yes, for example the Weierstraß function.
One can actually show that the set $A:= \{f \in C[0,1]; f$ has no right-derivative in any point in $[0,1)\}$ is dense in $C[0,1]$ and uncountable.
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