Mathematics 242 – Principles of Analysis Discussion 3: Integrability November 21, 2005

Background

On Friday, we introduced the upper and lower sums for a bounded function f on [a, b] relative to a partition P:

\begin{aligned}U(f, P) & = ∑_{i=1}^{n} M_{i}Δx_{i} \\ L(f, P) & = ∑_{i=1}^{n} m_{i}Δx_{i} \end{aligned}

where M_{i} = sup{f(x) : x ∈ [x_{i−1}, x_{i}]}, and m_{i} = inf{f(x) : x ∈ [x_{i−1}, x_{i}]}.

We showed that every lower sum is less than or equal to every upper sum:

L(f, P) ≤ U(f, P′)

for all partitions P, P′ of [a, b], and we introduced the terminology that f is integrable over [a, b] if

sup{L(f, P) : P all partitions of [a, b]} = inf{U(f, P) : P all partitions of [a, b]}.

(∗) Another way to say that f is integrable is to say that for any ϵ > 0, there exists some partition P of [a, b] such that

U(f, P) − L(f, P) < ϵ.

Using this alternate form of the definition, let’s consider some examples and properties of integrable functions.

Discussion Questions

A) By using the definition (∗) and considering regular partitions of [0, 2], show that f(x) = 1 − 6x^{2} is integrable over [0, 2] and compute ∫_{0}^{2} 1 − 6x^{2} dx.

B) Is the function

f(x) = \{ \begin{matrix}1 & if x is a rational number \\ 0 & if x is not a rational number \end{matrix}

integrable over [0, 1]? Why or why not? (Recall from the first part of the course that every interval of length > 0, no matter how small, contains both rational and irrational numbers.)

C) Here’s a somewhat surprising result. Suppose f is increasing on the whole interval [a, b]. Show that f must be integrable over [a, b] by considering the difference U(f, P_{n}) − L(f, P_{n}) for a regular partition P_{n} of [a, b] and letting n → ∞. (Hint: Δx = \frac{b−a}{n} can be factored out of every term in U(f, P_{n}) − L(f, P_{n}). When you do that, what happens in the sum?)

D) If c ∈ (a, b) and f is integrable over [a, c] and [c, b], then show f is integrable over [a, b] also, and

∫_{a}^{b} f(x) dx = ∫_{a}^{c} f(x) dx + ∫_{c}^{b} f(x) dx

Assignment

Writeups due Monday, November 28, no later than 5:00pm.