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Prove that $f(x)=\sum_{n=1}^\infty 1/2^n(\cos3^nx)$ is continuous but nowhere differentiable on $\mathbb{R}$.

I have proved the continuity part, but unable to do the second one. Thanks for any help.

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    I think this may be trickier than it looks at first sight. Weierstrass's example is $\sum_{n=1}^\infty a^n \cos(b^n \pi x)$ where $ab > 1 + 3 \pi/2$. The change of variables $t = \pi x$ gets rid of the first $\pi$, but you have only $ab = 3/2$, so Weierstrass's method is unlikely to work.2012-12-10
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    His method won't work directly, but the concept behind it will still apply, mainly due to the self-similarity issue this curve will have concerning differentiation. More specifically the curve can at most be differentiated at a countable number of points (and in this case that will not happen due to the lack of smoothness). For an exploration into fractal differentiation see: http://arxiv.org/pdf/1010.0881.pdf2012-12-10

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