Suppose that we have a function $f : [a,b] \to \Bbb{R} $ with countable discontinuities on $[a,b]$. Let call this set $D$
Take now the set$\space$$E =$ { $x \in [a,b] : f(x) \gt c $}$\space$ for a $c\in\Bbb{R}$.
My question is : Can a point in $D$ be in $E$? I ask this question because I was looking at this thread : A real valued function defined on a Borel subset of $\mathbb{R}$ with a countable number of discontinuities is Borel measurable
The second answer reads : “$$f^{-1}(B) = (f^{-1}(B) \cap E) \cup (f^{-1}(B)\cap D)= f|_E^{-1}(B) \cup (f^{-1}(B)\cap D)$$“
I don’t understand the point of the last union. I don’t know why it’s necessary to take into account the sets of discontinuities. It’s because the set $D$ can have points in the set $E$? Other than that, I don’t see why it’s necessary.
EDIT : If no point of $D$ is in $E$, why not simply do : $$ E = [a,b] \ \cap C$$ with $C$ the measurable set of continous points of $f$.