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# 8.4 The Ruler Function

8.38   Example (Ruler function.) We now present an example of an integrable function that is not monotonic on any interval of positive length. Define by

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.
This set resembles the markings giving fractions of an inch on a ruler, which motivates the name ruler function for . It is easy to see that is not monotonic on any interval of length . For each let be defined by

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
 (8.39) (8.40)

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 .

8.41   Exercise. A Let be the ruler function. We just gave a complicated proof that is integrable and . Explain why if you assume is integrable, then it is easy to show that .

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

Next: 8.5 Change of Scale Up: 8. Integrable Functions Previous: 8.3 A Non-integrable Function   Index
Ray Mayer 2007-09-07