Department of Mathematics

Colorado State University

Math 317: Homework 11

Due: Friday, December 6, 2019

Reading:

Read Section 34.

Exercises:

  1. Let the function be defined as,

    1. 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

    2. Is integrable on the interval ?

  2. 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.

  3. 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 .