# Harrison Chapman

## Math 317: Homework 11

Due: Friday, December 6, 2019

Exercises:

1. Let the function $$f$$ be defined as,

$f(x) = \left\{ \begin{array}{ll} x\;\; & x \in \mathbb Q \\ 0\;\; & x \in \mathbb R \setminus \mathbb Q \end{array} \right.$
1. 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

$U(f, Q \cup P_n) \ge \sum_{k=1}^{n}{ \frac{(k-1)b}{n} \left(\frac bn\right) }.$
2. Is $$f$$ integrable on the interval $$[0, b]$$?

2. 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.

3. 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 $$\int_a^b f = \lim U_n = \lim L_n$$.