XI. Integration #

Problems on upper and lower sums, criteria for Riemann integrability, and the basic order and mean-value properties of the integral.

Problem 1 #

Consider a function $f: [a,b] \to \mathbb{R}$ which may or may not be Riemann integrable. For any two partitions $P$ and $Q$ of the interval $[a,b]$, show that

$$L(f,P) \leq U(f,Q),$$

that is, any upper sum is greater than or equal to any lower sum for $f$.

Hint: Use a common refinement, that is, a partition that is both a refinement of $P$ and $Q$.

Solutions: Hint · Solution · Long solution

Problem 2 #

From class, we know that bounded continuous functions on a compact interval in $\mathbb{R}$ are Riemann integrable. The following exercise will show that a function can have one jump or removable discontinuity and still remain Riemann integrable. Consider an interval $[a,b] \subset \mathbb{R}$ and a point $c \in [a,b]$. Define a function $f : [a,b] \to \mathbb{R}$ by $f(x)=0$ unless $x=c$, when $f(c) = 1$. In other words, $f = \chi_{\{c\}}$, the indicator function of the set $\{c\}$.

(a) Show that $f$ is Riemann integrable on $[a,b]$.

(b) Conclude that any function which is continuous except possibly for a jump or a removable discontinuity at one point is Riemann integrable.

In fact, by induction one can extend this exercise to show that a finite number of such discontinuities do not affect Riemann integrability. Consequently, things like step functions are Riemann integrable as well. Can this requirement be relaxed even further?

The answer is “yes”, and in fact, by quite a bit. The complete answer was found by Henri Lebesgue in his doctoral thesis.

Therein, he also constructed what is known today as the Lebesgue integral, which has come to completely supersede the Riemann integral. But this is a story for Math 3603.

The complete answer involves the measure of a set. We will say a subset of $\mathbb{R}$ has measure zero if for every $\epsilon > 0$, it can be covered by a countable number of open intervals whose total length is less than $\epsilon$. It turns out that the rational numbers and the Cantor set both have measure zero. Here is Lebesgue’s observation:

Riemann–Lebesgue Theorem. Suppose that $f: [a,b] \to \mathbb{R}$ is a bounded function and let $D$ be the set of points where it is discontinuous. Then $f \in \mathcal{R}[a,b]$ if and only if $D$ has measure zero.

We will not have a chance to prove this in this class, so you will have to refrain from using this result in what follows!

Solutions: Hint · Solution · Long solution

Problem 3 #

Consider the function $f: \mathbb{R} \to \mathbb{R}$ defined by

$$f(x) = \begin{cases} \sin\left(\frac{1}{x}\right) & \text{if } x\in(0,1], \text{ and} \\[6pt] 0 & \text{if } x = 0. \end{cases}$$

It turns out that $f$ is not continuous at zero, but the discontinuity is neither a jump nor removable. (Points where a bounded function is discontinuous but the discontinuity is neither a jump nor removable are called essential discontinuities.) Nevertheless, prove that $f \in \mathcal{R}[0,1]$.

Solutions: Hint · Solution · Long solution

Problem 4 #

Consider two Riemann-integrable functions $f,g : [a,b] \to \mathbb{R}$. Show that if for all $x \in [a,b]$ we have $f(x) \leq g(x)$, then

$$\int_a^b f \leq \int_a^b g.$$

Solutions: Hint · Solution · Long solution

Problem 5 #

Suppose that $f: [a,b] \to \mathbb{R}$ is continuous and $f(x) \geq 0$ for all $x \in [a,b]$. Prove that if $\int_a^b f = 0$, then $f$ is the zero function.

Solutions: Hint · Solution · Long solution

Problem 6 #

Suppose that $f$ is a continuous real-valued function on the interval $[a,b]$. Prove that there exists a point $c \in [a,b]$ such that

$$\int_a^b f = f(c)\,(b-a).$$

This is the Mean Value Theorem for Integrals.

Solutions: Hint · Solution · Long solution