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.