Constructing explicit nowhere differentiable functions I was playing around with Desmos and graphed a few functions, for example $f(x)=\sum_{n=1}^{\infty}\frac{\sin(2^nx)}{n^2}$, $f(x)=\sum_{n=1}^{\infty}\frac{\sin(n!x)}{n^4}$, and $f(x)=\sum_{n=1}^{\infty}\frac{\sin(nx)}{n^2}$. Notice that all of these functions converge absolutely. 
The graphs of these functions led me to a conjecture: if $f,g:\mathbb{N}\rightarrow \mathbb{N}$ satisfy $\lim_{n\rightarrow \infty}f(n)/g(n)=\infty$ and that $h(x):=\sum_{n=1}^{\infty}\frac{\sin(f(n)x)}{g(n)}$ is well-defined, then $h(x)$ is nowhere differentiable. 
My question is whether if this is a valid conjecture. In either case, are the hypotheses too weak/too strong? Also, are there more general results than the one I have above? 
 A: Partial but stronger result for lacunary series -
Let $S(x)=\sum_{k \ge 1}c_k \sin \lambda_kx, C(x)=\sum_{k \ge 1}c_k \cos \lambda_kx$ where $\lambda_m \to \infty $ and there is some fixed $q>1$ with $\lambda_{n+1} >q\lambda_n, n >n_0$. We assume that $S,C$ are Fourier series of integrable functions (for example $\sum{|c_k|^2} < \infty$ is sufficient as then $S,C$ represent $L^2$ functions and actually - though that is highly non-trivial- necessary too)
For example $h(x)=\sum_{n=1}^{\infty}\frac{\sin(2^nx)}{n^2}, h_1(x)=\sum_{n=1}^{\infty}\frac{\sin(n!x)}{n^4}$ are lacunary but something like $R(x)=\sum_{n=1}^{\infty}\frac{\sin(n^2x)}{n^2}$ is not lacunary.
Then all such $S,C$ are differentiable at a point iff $c_k\lambda_k \to 0, k \to \infty$ and then they are differentiable on a dense set.
In particular for the OP, in the lacunary condition $f(n+1)>qf(n), n >n_0, q>1$ it is enough to have $\limsup \frac{f(n)}{g(n)} >0$ for the respective series to be nowhere differentiable when say $\sum{|g(n)|} < \infty$ so they are absolutely convergent Fourier series (continuos etc)
For the series $R(x)=\sum_{n=1}^{\infty}\frac{\sin(n^2x)}{n^2}$ which is not lacunary, the result is not true as the function is differentiable at all $x=r\pi$ where $r$ is a rational which can be written as the ratio of two odd integers (and nowhere else); as in this case, $f(n)/g(n)=1$, it is not quite a counterexample for the OP but it shows that even though the formal derivative is not-convergent anywhere, the series can still be differentiable.
The result for lacunary Fourier series which implies immediately the non-differentiability of any point of Weierstrass famous function $\sum {2^{-n}\cos 2^nx}$ is not that hard to prove and a good reference for it is Katznelson An Introduction to harmonic Analysis Chapter 5
The result for the Riemann function $R$ (sometimes called "Riemann's other function" though it is nowhere as famous as the RZ of course) is fairly hard and was only completed in the 1960's (anecdotes have it that Riemann thought this is an example of a continuous nondifferentiable function and ~1860 gave Weierstrass the task to prove that and when Weierstrass couldn't - it's not true after all as it is differentiable at $\pi$ for example - Weierstrass came up with his famous function). A nice presentation of this is in the wonderful Twelve Landmarks of Twentieth-Century Analysis by D Choimet and H Queffelec chapter 7.
