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I was wondering if it there is a function $f:\mathbb{R} \rightarrow \mathbb{R}$ that is differentiable everywhere, but with a derivitive that is nowhere differentiable.

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You could integrate the Weierstrass function. –  Dylan Moreland Feb 25 '12 at 6:05

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up vote 5 down vote accepted

Look up your favorite example of a continuous-but-nowhere differentiable function, then integrate it. By the fundamental theorem of calculus you'll get the original function back when you differentiate.

http://en.wikipedia.org/wiki/Weierstrass_function

http://www.proofwiki.org/wiki/Continuous_Nowhere_Differentiable_Function

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Yes, there is. Let $g$ be the Weierstrass function, and put $$ f(x)=\int_0^x g(t)dt. $$ Then $f$ is everywhere differentiable because it is an integral of a continuous function. But $f'=g$, which is nowhere differentiable.

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Beaten to the punch! +1 –  Alex Becker Feb 25 '12 at 6:07
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Oh, right. That's pretty cool. Thanks –  Chris Dugale Feb 25 '12 at 17:15

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