Math 104 Assignment 11

1. A pathological function

   Let $f : [0,1] \to \R$ be defined as follows: \[ f(x) = \begin{cases} 0 & \text{ if } x \notin \Q \\ 1 & \text{ if } x = 0 \\ \frac1q & \text{ if } x = \frac{p}{q} \text{ with } p \in \Z, q \in \N, \text{ and } p, q \text{ have no common factor}.\end{cases}\] (For example, $f(0) = f(1) = 1$, $f(1/2) = 1/2$, $f(1/4) = f(3/4) = 1/4$.)

   Prove that $f$ is integrable, and find its integral.

2. Integrals are insensitive to individual points

   1. This week, we will see that continuous functions are integrable. Use this to prove that if $f : [a, b] \to \R$ is bounded and continuous except at finitely many points, then $f$ is integrable.
   2. Suppose that $f : [a, b] \to \R$ is integrable, and $g : [a, b] \to \R$ is such that $f(x) = g(x)$ for all but finitely many $x \in [a, b]$. Show that $g$ is integrable, and \[\int_a^b f(x)\,dx = \int_a^b g(x)\,dx.\]

3. Properties of integrals

   1. Let $f : [0, 1] \to \R_{\geq 0}$ be continuous. Show that $f = 0$ (that is, $f$ is the constant function $0$) if and only if \[\int_0^1 f(t)\,dt = 0.\]
   2. Show that if the assumption of continuity is dropped in Part A, it is no longer true.
   3. Suppose that $f, g : [a, b] \to \R$ are such that $f(t) \leq g(t)$ for all $t \in \R$. Show that $U(f) \leq U(g)$. (Note that, as a consequence, if $f$ and $g$ are integrable then $\int_a^b f(t)\,dt \leq \int_a^b g(t)\,dt$.)
   4. Suppose that $f : [a, b] \to \R$ is monotonically increasing, i.e., if $a \leq t \lt s \leq b$ then $f(t) \leq f(s)$. Prove that $f$ is integrable.

4. An integral of integrals

   Suppose that $f : [0, 1] \times [0, 1] \to \R$ is continuous. Notice that for each $y_0 \in [0, 1]$, we have a function \begin{align*} [0, 1] &\longrightarrow \R \\ x &\longmapsto f(x, y_0). \end{align*} Each of these functions is continuous, and so integrable; this can be checked directly from the definition, or by using the sequential characterisation of continuity.

   1. Prove that for any $\epsilon \gt 0$ there is a $\delta \gt 0$ so that if $\abs{y - z} \lt \delta$, we have \[ \sup_{x \in [0, 1]} \abs{f(x, y) - f(x, z)} \lt \epsilon. \]
   2. Show that the function \begin{align*} I: [0, 1] &\longrightarrow \R \\ y &\longmapsto \int_0^1 f(x, y) \,dx \end{align*} is continuous, and therefore integrable.

Here are some further problems to think about out of interest. You do not need to attempt them, nor should you submit them with the assignment.

• Not definitions

  1. We say a function $f : X \to Y$ is funtinuous at $p \in X$ if for every $\epsilon \gt 0$ there is some $\delta \gt 0$ so that for every $x, y \in X$ with $d_X(x, y) \lt \delta$ we have $d_Y(f(x), f(y)) \lt \epsilon$.
     1. Are all functions funtinuous?
     2. Are all continuous functions funtinuous?
     3. Are all funtinuous functions continuous?
     4. Does a funtinuous function exist?
  2. We say a function $f : X \to Y$ is protinuous if for every $x_0 \in X$, $\forall\delta \gt 0$, for all $x \in X$ with $d_X(x, x_0) \lt \delta$ there exists $\epsilon \gt 0$ so that $d_Y(f(x), f(x_0)) \lt \epsilon$.
     1. Are all functions protinuous?
     2. Are all continuous functions protinuous?
     3. Are all protinuous functions continuous?
     4. Does a protinuous function exist?
  3. We say a function $f : X \to Y$ is antitinuous at a point $p \in X$ if for every $\epsilon \gt 0$ and all $x \in X$, there exists $\delta \gt 0$ so that $d_X(x, p) \lt \delta$ implies $d_Y(f(x), f(p)) \lt \epsilon$.
     1. Are all functions antitinuous?
     2. Are all continuous functions antitinuous?
     3. Are all antitinuous functions continuous?
     4. Does a antitinuous function exist?
  4. We say a function $f : X \to Y$ is continually uniformous if for all $x, y \in X$ and all $\epsilon \gt 0$ there is a $\delta \gt 0$ so that $d_X(x, y) \lt \delta$ implies $d_Y(f(x), f(y)) \lt \epsilon$.
     1. Are all functions continually uniformous?
     2. Are all uniformly continuous functions continually uniformous?
     3. Are all continually uniformous functions uniformly continuous?
     4. Does a continually uniformous function exist?
  5. We say a function $f : X \to Y$ is continuously uniform if for every $\epsilon \gt 0$ there is a $\delta \gt 0$ so that for all $x, y \in X$, $d_Y(f(x), f(y)) \lt \epsilon$ implies $d_X(x, y) \lt \delta$.
     1. Are all functions continuously uniform?
     2. Are all uniformly continuous functions continuously uniform?
     3. Are all continuously uniform functions uniformly continuous?
     4. Does a continuously uniform function exist?

• Integrals of integrals

  Let $f : [0, 1]\times[0, 1] \to \R$.

  1. Suppose rather than assuming that $f$ is continuous, we merely assume that for each $y \in [0, 1]$, the function $x \mapsto f(x, y)$ is integrable. Is \[ y \mapsto \int_0^1 f(x, y)\,dx \] continuous? Integrable?
  2. Suppose $f$ is continuous. Problem 4 allows us to define the integral \[\int_0^1 \int_0^1 f(x, y)\,dx\,dy,\] and a similar argument shows that \[\int_0^1 \int_0^1 f(x, y)\,dy\,dx\] exists as well. Must these two quantities be equal?
  3. Suppose rather than assuming that $f$ is continuous, we merely assume that for each $y \in [0, 1]$, the functions \[ x \mapsto f(x, y) \hspace{2cm}\text{and}\hspace{2cm} x \mapsto f(y, x) \] are both integrable. Must \[ y \mapsto \int_0^1 f(x, y)\,dx \hspace{2cm}\text{and}\hspace{2cm} y \mapsto \int_0^1 f(y, x)\,dx\] be integrable?
  4. Suppose now that $f$ is such that \[ x \mapsto f(x, y) \hspace{2cm}\text{and}\hspace{2cm} x \mapsto f(y, x) \] are integrable for each $y \in [0, 1]$, and that \[ y \mapsto \int_0^1 f(x, y)\,dx \hspace{2cm}\text{and}\hspace{2cm} y \mapsto \int_0^1 f(y, x)\,dx\] are integrable. Must \[ \int_0^1 \int_0^1 f(x, y)\,dx\,dy = \int_0^1 \int_0^1 f(x, y)\,dy\,dx\text{?} \]