My book gave this as an excercise and I did it with a method similar to one given as an example, but I am curious of this alternate method will work. Forgive me if the proof is incoherent or just is wrong, I haven't slept in two days (long story).
Question: Let . Is on and if so what is the value of
Notation: For my sake let
Answer: By definition we must show that there exists a partition of such that . Now it is clear that since on any interval there exists values of such that , so consequently which in turn implies that for any partiton . This realization enables to prove by proving that .
To do this we must merely construct our partition carefully. Before we do this let me establish some "new" notation.
Define as being a neighborhood of radius around
Now construct a partition of as follows: Let a partition of posses the following charcteristics , , and . Now it is clear that we must have .
So now it is obvious that we can make the in as big as we want by making as small as we want, so we can make similarly as small as we want. Finally we can make as small as we want.
Now let be a number such that . Choose a partition of such that . Let , then
This proves integrability of . Now to find the value just note that since it is integrable that
Sorry if that is really far off.