Read Section 34.
Let the function be defined as,
Compute the upper and lower Darboux integrals for on the interval .
Hint. Here’s how you can show that : For any partition , if , explain why
Is integrable on the interval ?
Let be a bounded function on . Suppose that is integrable. Does it follow that is also integrable? If so, prove it. If not, provide a counterexample.
Let be a bounded function on . Suppose there exist sequences and of upper and lower Darboux sums for such that . Show is integrable on and that .