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.
Differentiable function with nowhere differentiable derivitive
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real-analysis
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6You could integrate the [Weierstrass function](http://en.wikipedia.org/wiki/Weierstrass_function). – 2012-02-25
2 Answers
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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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1Oh, right. That's pretty cool. Thanks – 2012-02-25