On the differentiability of increasing functions

Question

I'm trying to follow the proof of this website http://mathonline.wikidot.com/lebesgue-s-theorem-for-the-differentiability-of-monotone-fun about the differentiability a.e. of increasing functions. Although, there's something I don't understand. It says:

"Now note that for each $k\in\{1,2,...,n\}$ we have that $E\cap[c_k,d_k]\subseteq\{x\in[c_k,d_k]:\overline{D}f(x)\geq|\alpha|\}$".

I don't know how to obtain that absolute value of $\alpha$. The only thing I know about that set is the following:

$$E\cap[c_k,d_k]=\{x\in[c_k,d_k]:\underline{D}f(x)<\beta<\alpha<\overline{D}f(x)<+\infty\}.$$

Thanks in advance.

Answer 1

Because $f$ is increasing, all difference quotients $$\dfrac{f(x+h) - f(x)}h \ge 0, h \ne 0$$ Thus $\underline Df(x) \ge 0$ for all $x$. The only way for $E \cap [c_k, d_k]$ to be non-empty is for $\alpha \ge 0$, so $|\alpha| = \alpha$ and since you already know that $\overline Df(x)>\alpha$, the inclusion follows.

And if $E \cap [c_k, d_k]$ is empty, then it is a subset of every set.