How can I show $F$ is not monotonic in any subinterval? I'm trying to solve Problem 1(b) of [Fourier Analysis] by Stein. The problem is as follows:
Choose a countable dense sequence ${r_n}$ in $[0,1]$. Consider
$\displaystyle{F(x)=\sum_{n=1}^\infty3^{-n}g(x-r_n)}$,
where $g(x)=\sin(1/x)$ when $x\neq0$, and $g(0)=0$. Then $F$ fails to be monotonic in any subinterval of $[0,1]$.
The Author gives as a hint $3^{-k}>\sum_{n>k}3^{-n}$. But I cannot figure out how to use this hint. Since $F$ is a superposition of highly oscillating functions, it seems that $F$ is not monotonic.

It is sufficient to show that $F(x)$ is almost nowhere differentiable, since 
a theorem of Lebesgue gives that a monotonic function over an interval $I$ is almost everywhere differentiable over $I$.
