# Determine conditions for $a,b>0$ such that $f(x)=\sum b^n\sin(a^nx)$ be continuous but nowhere differentiable in $\mathbb{R}$

Determine conditions for $a,b>0$ such that $f(x)=\sum b^n\sin(a^nx)$ be continuous but nowhere differentiable in $\mathbb{R}$.

Attempt: If $0<b<1$ the function is clearly continuous by M-test and uniform convergence of continuous functions.

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According to Gelbaum and Olmsted, Counterexamples in Analysis, page 39, $\sum_0^{\infty}b^n\cos(a^n\pi x)$ is continuous and nowhere differentiable if $b$ is an odd integer, $0\lt a\lt1$, and $ab\gt1+(3\pi/2)$, result due to Weierstrass. en.wikipedia.org/wiki/Weierstrass_function says all you need is $0\lt a\lt1$, $ab\ge1$, citing a paper of Hardy. – Gerry Myerson Nov 27 '12 at 1:48
@GerryMyerson Can you post your comment as an answer? I dont want that this answer appears in unanswered list. – Gastón Burrull Apr 2 '13 at 21:01

According to Gelbaum and Olmsted, Counterexamples in Analysis, page 39, $$\sum_0^{\infty}b^n\cos(a^n\pi x)$$ is continuous and nowhere differentiable if $b$ is an odd integer, $0\lt a\lt1$, and $ab\gt1+(3/2)\pi$, a result due to Weierstrass. Wikipedia says all you need is $0\lt a\lt1$, $ab\ge1$, citing a paper of Hardy.
EDIT: As @zyx notes in a comment, $a$ and $b$ have been switched. I have quoted it exactly as I found it in the book, but that's no excuse --- I should have noticed the error.
$a$ and $b$ are switched. – zyx Apr 4 '13 at 1:54