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.

Elevating comment to answer, at request of OP: 
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. 
