Let $E$ have measure zero contained in the open interval $(a,b)$. In a previous problem I showed that there is a countable collection of open intervals, $\{(c_k,d_k)\}_k$, contained in $(a,b)$ for which each point in $E$ is contained in infinitely many intervals of the collection and $\sum_k |(c_k,d_k)|=\sum_kd_k-c_k<\infty.$ Define $f(x)=\sum_k|(c_k,d_k)\cap(-\infty,x)|$ for $x \in (a,b)$. Show $f$ is increasing and fails to be differentiable for every point in $E$.
The absolute value bars mean measure of the interval. I have already shown $f$ was increasing by showing that if $a \le u <v \le b$ then $$\begin{align} f(v)-f(u) &=\sum_k |(c_k,d_k)\cap [(-\infty,u)\cup[u,v)] |-\sum_k|(c_k,d_k)\cap (-\infty,u)|\\&=\sum_k|(c_k,d_k)\cap [u,v)] |\ge 0 \end{align}$$
Let $D^+f(x)=\lim_{h\rightarrow 0}\left[\sup_{0<|t|\le h}\dfrac{f(x+t)-f(x)}{t} \right]$ and $D^-f(x)=\lim_{h\rightarrow 0}\left[\inf_{0<|t|\le h}\dfrac{f(x+t)-f(x)}{t} \right]$
To show f is not differentiable at any point in $E$ then I need to show that $D^+f(x) \not= D^-f(x)$ for every $x \in E$. This is where I am having trouble. First from using what I did above to show that $f$ is increasing I found that for $t>0$
$$ A_t(f(x))= \dfrac{f(x+t)-f(x)}{t}=\dfrac{1}{t}\sum_{k=1}^{\infty}|(c_k,d_k)\cap[x,x+t)| $$
So at this point what we know is that if $x\in E$ then $x$ belongs to $(c_k,d_k)$ for infinitely many $k$ and $\sum_{k=1}^{\infty}|(c_k,d_k)\cap[x,x+t)|$ converges since $\sum_{k=1}^{\infty}|(c_k,d_k)|<\infty$. But Im not sure how to find the $\sup_{0<|t|\le h}$ and $\inf_{0<|t|\le h}$ of $A_t(f(x))$ I think that $\inf_{0<|t|\le h}A_t(f(x))=0$ and $\sup_{0<|t|\le h}A_t(f(x))>0$ but I dont know how to show it.
Any help is appreciated, thanks