I have to find a derivative in a distributional sense of the following function (known as Cantor's singular function) $$f(x)=\left\{ \begin{array}{l l l l l} 0, & \quad\text{$ x\leq 0 $}\\ 1, & \quad\text{$ x\geq 1 $} \\ \frac{1}{2}, & \quad\text{$ x\in[\frac{1}{3},\frac{2}{3}] $ }\\ \frac{1}{4}, & \quad\text{$ x\in[\frac{1}{9},\frac{2}{9}] $}\\ \frac{3}{4}, & \quad\text{$ x\in[\frac{7}{9},\frac{8}{9}] $}\\ .\\ .\\ . \end{array} \right.$$
I am not sure if my solution is correct:
Function f is constant on each open interval (($-\infty,0)$, $(1,\infty)$, $(\frac{1}{3},\frac{2}{3})$, ...), so $f'(x)=0$ a.e. Therefore, $<f'(x),\phi(x)>=0$, for any test function $\phi(x)$.
Is my solution correct? Thanks in advance.