This formula defines uniquely: If where and are odd, then . (If , we get , which says that an even number is odd.) The set under the graph of is shown in the figure.

We have seen that is integrable on any interval and . Now define a sequence of functions by

Each function is integrable with integral and

I will now show that is integrable.

Let be a partition-sample sequence for . I'll show that .

Let be a generic element in
.
Observe that if
then

Hence by the Archimedian property, we can choose so that . Then

Now since for all , we have

Since is integrable and , we have so there is an such that for all . By equation (8.40) we have

Hence , and hence is integrable and .

Also show that if you assume that the non-integrable function in equation (8.37) is integrable then it is easy to show that .