Let $f,g:\mathbb R\to\mathbb R$ be two continuous but nowhere differentiable functions. By the Denjoy–Young–Saks theorem for almost every point $x_0\in\mathbb R$ one has 
$$
\limsup\limits_{x\to x_0}\frac{|f(x)-f(x_0)|}{|x-x_0|}=+\infty.
$$

Is then $f\circ g$ still nowhere differentiable? It seems possible that the high frequency parts of $g$ lie in the low frequency parts of $f$, which might lead to a cancellation of the oscillation. 

I am trying to prove nondifferentiability through the following: Fix $x_0$ and consider 
$$
I_{f,x_0}^k=\{x\in\mathbb R:|f(x)-f(x_0)|\ge k|x-x_0|\},
$$
the set of points outside a double-sided cone with slope $k$. Is that true that
$$
\lim\limits_{\delta\to0}|I_{f,x_0}^k\cap[x_0-\delta,x_0+\delta]|/(2\delta)=1?
$$
If yes then we know that “most” of the points near $x_0$ have values relatively far away from $f(x_0)$, and so there will not be much frequency cancellation.