Reading:
Read Section 34.
Exercises:
Let the function \(f\) be defined as,
\[f(x) = \left\{ \begin{array}{ll} x\;\; & x \in \mathbb Q \\ 0\;\; & x \in \mathbb R \setminus \mathbb Q \end{array} \right.\]Compute the upper and lower Darboux integrals for \(f\) on the interval \([0,b]\).
Hint. Here’s how you can show that \(U(f) \ge b^2/2\): For any partition \(Q\), if \(P_n = \{\cdots \frac{kb}n \cdots\}\), explain why
\[U(f, Q \cup P_n) \ge \sum_{k=1}^{n}{ \frac{(k-1)b}{n} \left(\frac bn\right) }.\]Is \(f\) integrable on the interval \([0, b]\)?
Let \(f\) be a bounded function on \([a, b]\). Suppose that \(f^2\) is integrable. Does it follow that \(f\) is also integrable? If so, prove it. If not, provide a counterexample.
Let \(f\) be a bounded function on \([a, b]\). Suppose there exist sequences \((U_n)\) and \((L_n)\) of upper and lower Darboux sums for \(f\) such that \(\lim(U_n - L_n) = 0\). Show \(f\) is integrable on \([a,b]\) and that \(\int_a^b f = \lim U_n = \lim L_n\).