Homework 17: The Riemann Integral

Instructions. Assignments should be stapled and written clearly and legibly. For all problems, you can assume that if a function \(f\) is continuous on \([a,b]\) then it is integrable on \( [a,b]\). I recommend that you solve Problem 4 before solving the Section 7.2 textbook problems. Problems 5 and 6 are optional.

1. Section 7.1, #15.
2. Section 7.2, #6, 11, 12.
3. Let \(f(x)=x^2-x\) and \(P=\left\{0,\frac{1}{2},1,\frac{3}{2},2\right\}\). Find \(U(f,P)\) and \(L(f,P)\).
4. Suppose that \(f:[a,b]\to \mathbb{R}\) is continuous, \(f(x)\geq 0\) for all \(x\in [a,b]\), and \(f(x)\gt 0\) for at least one value \(c\in [a,b]\). Using definitions, prove that \(\int_a^b f \gt 0\). (You may assume that \(f\) is integrable.)
5. (Challenge) Consider the function \(f:[0,2]\to \mathbb{R}\) given by \begin{equation*} f(x)= \begin{cases} 0 & \text{if }x = \frac{1}{n}\text{ for some }n\in \mathbb{N}\\ 1 & \text{otherwise} \end{cases} \end{equation*} Prove that \(f\) is integrable and find \(\int_0^2 f\).
6. (Challenge) Let \(h\) be Thomae's function of Homework 16, Problem 6. In this problem, you will prove that \(h\) is integrable.
   1. Prove that \(L(h,P)=0\) for any partition \(P\) of \([0,2]\).
   2. Let \(\epsilon \gt 0\). Let \(S=\{x\in [0,2]:h(x)\gt \epsilon/4\}\). Determine whether \(S\) is finite or infinite.
   3. Explain how to construct a partition \(P\) of \([0,2]\) for which \(U(h,P) \lt \epsilon\). Prove that your partition works.
   4. Complete the proof.