Following is from Bruckner's Analysis book which I after lots of time try couldn't understand two parts :
My questions :
1- Even though I understood the proof of Theorem 1.18 completely, but still how the author concludes from that there must be $(J,E_n \subset J)$ such that $\text{Cl}(E_n)=J$?
2- In the last lines how it is concluded that $J \cap F(\epsilon) = \emptyset$? Elements of oscillation are defined for difference of the same $f$ at two points but the elements of $J$ are difference of two functions for one same point!
The interval $I$ cannot be a countable sum of nowhere dense subsets, so at least one of the sets $E_n$ must be not-nowhere dense. This means (from the definition) that its closure has non-empty interior, so it contains an open interval $J$. In particular, ${\rm Cl}(E_n\cap J) = J$.
Assuming that the interval $J$ is open and $x\in J$ is fixed, we have $$|f(x) - f_n(x)| < \epsilon/2,$$ but also $$|f(y) - f_n(y)|<\epsilon/2$$ whenever $y \in J$. If we take $\delta$ small enough, then $(x-\delta,x+\delta)\subset J$ and thus for every $y\in (x-\delta,x+\delta)$ $$|f(y)-f(x)| \leq |f(y) - f_n(y)| + |f(x) - f_n(x)| + |f_n(y) - f_n(x)| < \epsilon/2 + \epsilon/2 + c(\delta),$$ where $c(\delta) \to 0$ as $\delta\to 0^+$, because $f_n$ is continuous. Thus, by letting $\delta\to 0^+$ we get that $x\notin F(\epsilon)$.
It is instructive to realize that in the definition of oscillation we can as well take the limit with $\delta\to 0^+$, because the quantity under $\inf$ decreases (as $\delta$ decreases) and is bounded from below.