I got stuck trying to prove that:
$T(x)$ is nowhere monotonic
$T(x)$ is nowhere differentiable
$$f:\mathbb R\to\mathbb R\ \ \ \ \ \ \ \ \ f(x)= |x|\quad \text{for} \quad\frac{-1}{2}\le x\le \frac{1}{2} \quad \text{and $f$ is periodic with period $1$}$$
$$\text{for }n =0,1,2,...\quad f_n(x):=\frac{f(4^n x)}{4^n}$$
$$T(x):=\sum_{n=0}^\infty f_n (x) = \sum_{n=0}^\infty \left(\frac{f(4^n x)}{4^n}\right)$$