Let be bounded on . Show that if the set of discontinuous points of has content zero, then is integrable.
I am trying do a second semester analysis self study and I am pretty stuck on a few major concepts of integration. I would appriciate some help with this one. If any thing, I just need a place to get started.
Here is what I have set up so far. I am given:
has content zero and is bounded on . Thus, and (where, represents open sets). Moreover, there exists an satistying for all .
My goal is to show there exists a partition such that for any choice of
Since the goal begins with a universal quantifier I let be arbitrary. Now, I need to show the existance of such a partition.
As you can see I do not have much. I am not sure how to get started.
Q1: Does the set of open intervals whos union contain the set D have to be nested?
Q2: Can I assume is compact to create a finite subcover and use that somehow?