How to prove fractal is differentiable at no point?

Question

this is a problem from Elementary Classical Analysis. Author constructs a function which is continuous everywhere but is differentiable at no point, and i don't how to prove the second result. Here is the problem: Construct a function $g(x)$ by letting $g(x)=|x|$ if $x\in [\frac{-1}{2},\frac{1}{2}]$ and extending g so that it becomes periodic. Define $\sum_{n=1}^\infty \frac{g(4^{n-1}x)}{4^{n-1}}$ I know could use Weierstrass M test to prove the first result, but I can't prove that this function is differentiable at no point. Could someone give me some hints?