Understanding steps in a proof.

Question

A part of the proof is given below:

My questions are:

1. I do not understand why "Since $f'(x) = 0$, we can find $y_{x} > 0$ such that we have both $[x,y_{x}] \subseteq (a,b)$ ".

   Could anyone explain this step for me please?

2. Why the set $\mathcal{B}$ is a vitali cover of $E$? Where is the required condition on the $\varepsilon $ i.e. $\ell[x,y_{x}] < \varepsilon $ for every $\varepsilon >0.$

   Could anyone explain this for me, please?

Answer 1

First question : just apply definition of $f'(x)$.

Second question: Vitali cover means for every $\epsilon >0$ there exists a member of the cover with length less than $\epsilon$. [ Not that every member of the cover has length less than $\epsilon$ ]. Since $[x,y]$ is included in this cover for every $y$ with $x <y <y_x$ this property is obviously satisfied.