# Harrison Chapman

## Math 317: Homework 11

Due: Friday, December 6, 2019

Reading:

Read Section 34.

Exercises:

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

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

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