Math 104 Assignment 11

1. Inverses

   Suppose that $f : (a, b) \to \R$ is differentiable with $f'(x) \gt 0$ for all $x \in (a, b)$.

   1. Prove that $f$ is strictly monotonically increasing, i.e., if $a \lt x \lt y \lt b$ then $f(x) \lt f(y)$.
   2. Note that $f$ is one-to-one, and so has an inverse function $g : f((a, b)) \to (a, b)$. Show that $g$ is continuous.
   3. Show that for all $x \in (a, b)$, we have $f(x) \in f((a, b))^\circ$ (the interior of $f((a,b))$).
   4. We saw that if $g$ is differentiable at $f(x)$, it must be the case that \[g'(f(x)) = \frac1{f'(x)}.\] Verify that $g$ is differentiable at $f(x)$.

2. 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.

3. Integrals are insensitive to individual points

   Next week, we will prove that linear combinations of integrable functions are integrable, and that when $f, g : [a, b] \to \R$ are integrable and $\lambda \in \R$, we have \[\int_a^b f(x) + \lambda g(x) \,dx = \int_a^b f(x)\,dx + \lambda \int_a^b g(x)\,dx.\] We will also prove that continuous functions are integrable. These facts may come in useful in this problem.

   1. Suppose that $f : [a, b] \to \R$ is non-zero at only finitely many points. Prove that $f$ is integrable, with \[\int_a^b f(t)\,dt = 0.\]
   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 \[\int_a^b f(s)\,ds = \int_a^b g(y)\,dy.\]
   3. Suppose that $f : [a, b] \to \R$ and $g : [b, c] \to \R$ are integrable, and define \begin{align*} h : [a, c] &\longrightarrow \R \\ t &\longmapsto \begin{cases} f(t) & \text{ if } t \in [a, b] \\ g(t) &\text{ otherwise.}\end{cases}\end{align*} Prove that $h$ is integrable, with \[\int_a^c h(t)\,dt = \int_a^b f(t)\,dt + \int_b^c g(x)\,dx.\]
   4. Suppose that $\varphi : [a, b] \to \R$ is bounded, and continuous at all but finitely many points in $[a, b]$. Prove that $\varphi$ is integrable.

4. 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 $\int_a^b f(t)\,dt \leq \int_a^b g(t)\,dt$.)

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.

• Monotonic functions

  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.

• 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.

• 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?

• Positive derivative at a point

  Suppose that $f : \R \to \R$ is differentiable, and $f'(0) \gt 0$.

  1. Show that if $f'$ is continuous at $0$, then there is some $\delta\gt0$ so that $f$ is monotonically increasing on $(-\delta, \delta)$.
  2. Show by example that this need not be the case if $f'$ is not continuous at $0$.

• 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 \[ \int_0^1 \int_0^1 f(x, y)\,dx\,dy = \int_0^1 \int_0^1 f(x, y)\,dy\,dx\text{?} \]

• Bounded derivatives

  Suppose that $f : \R \to \R$ is differentiable with bounded derivative. Prove that $f$ is uniformly continuous.

• Other perspectives on integration

  Let $a \lt b$ be real numbers, and suppose $f : [a, b] \to \R$ is bounded.

  $$ \newcommand{\cP}{\mathcal{P}} \newcommand{\cQ}{\mathcal{Q}} \newcommand{\cT}{\mathcal{T}} \newcommand{\sP}{\mathscr{P}} \newcommand{\sT}{\mathscr{T}} $$

  1. A tagged partition of $[a, b]$ is a pair $(\cP, \cT)$ where $\cP = \set{x_0 \lt x_1 \lt \cdots \lt x_n}$ is a partition of $[a, b]$, and $\cT = \set{t_1, \ldots, t_n}$ is a finite set so that $t_i \in [x_{i-1}, x_i]$. The Riemann sum of $f$ associated to $(\cP, \cT)$ is the number \[R(f, \cP, \cT) = \sum_{i=1}^n f(t_i) (x_i - x_{i-1}).\]

     We say that $f$ is Riemann integrable with Riemann integral $I$ if for any $\epsilon \gt 0$ there is a partition $\cP$ of $[a, b]$ so that for any tagged partition $(\cQ, \cT)$ where $\cQ$ is a refinement of $\cT$, we have \[\abs{R(f, \cQ, \cT) - I} \lt \epsilon.\]

     Prove that a function is Riemann integrable if and only if it is integrable (in the sense from class, that $L(f) = U(f)$), and in this case its Riemann integral is $\int_a^b f(t)\,dt$.

  A partially ordered set $(\Lambda, \leq)$ is called a directed set if every pair of elements has an upper bound; i.e., if for any $\alpha, \beta \in \Lambda$ there is some $\gamma \in \Lambda$ so that $\alpha \leq \gamma$ and $\beta \leq \gamma$. If $X$ is a set, a net along $\Lambda$ in $X$ is a function $\Lambda \to X$; we will sometimes denote it by writing $(x_\alpha)_{\alpha \in \Lambda}$ for the net defined by $\alpha \mapsto x_\alpha$. Nets should be though of as generalizations of sequences; in particular, $\N$ is a directed set and a sequence in $X$ is just a net in $X$ along $\N$.

  Suppose now that $X$ is a metric space, $\Lambda$ is a directed set, and $(x_\alpha)_{\alpha \in \Lambda}$ is a net. If $x \in X$, a neighbourhood of $x$ is an open set $U \subseteq X$ with $x \in U$. We say that the net converges to a limit $x \in X$ if for every neighbourhood $U$ of $x$, there is some $\lambda \in \Lambda$ so that whenever $\alpha \in \Lambda$ with $\alpha \geq \lambda$, we have $x_\alpha \in U$. In this case, we write \[\lim_{\alpha\in\Lambda} x_\alpha = x.\] (It turns out that nets are to arbitrary topological spaces what sequences are to metric spaces; sequences are in a certain sense not rich enough to capture the entire structure of a topology which does not come from a metric, but nets can. For example, its possible for a subset of a topological space to contain the limit of every convergent sequence with terms drawn from that set, but not be closed; such a set will contain the terms of some convergent net but not its limit. Likewise it's possible that a function commutes with the limit of every sequence but is not continuous, because it fails to commute with the limit of some net.)

  2. Prove that if $\Lambda$ has a maximum element, then every net along $\Lambda$ converges.
  3. Prove that if $(x_\alpha)_{\alpha\in\Lambda}$ and $(y_\alpha)_{\alpha\in\Lambda}$ are convergent nets in $\R$, then $(x_\alpha + y_\alpha)_{\alpha \in \Lambda}$ is convergent and \[\lim_{\alpha \in \Lambda} x_\alpha + y_\alpha = \lim_{\alpha \in \Lambda} x_\alpha + \lim_{\alpha\in\Lambda} y_\alpha.\] Prove a similar statement about multiplication and division of convergent nets (non-zero in the case of the quotient), and that constant nets converge.
  4. Let $\sP$ be the set of partitions of $[a, b]$. For $\cP_1, \cP_2 \in \sP$, let us write $\cP_1 \leq \cP_2$ if and only if $\cP_2$ is a refinement of $\cP_1$. Show that $\leq$ is a partial order on $\sP$, which makes it into a directed set.
  5. Show that \[ U(f) = \lim_{\cP \in \sP} U(f, \cP) \hspace{2cm}\text{and}\hspace{2cm} L(f) = \lim_{\cP \in \sP} L(f, \cP). \] Conclude that $f$ is integrable if and only if \[\lim_{\cP \in \sP} U(f, \cP) - L(f, \cP) = 0.\]
  6. Prove that the net $\paren{U(f, \cP) - L(f, \cP)}_{\cP \in \sP}$ is monotonic in the sense that if $\cP \leq \cQ$ then $U(f, \cQ) - L(f, \cQ) \leq U(f, \cP) - L(f, \cP)$. Use this to conclude that $f$ is integrable if and only if there is a partition $\cP \in \sP$ for which \[U(f, \cP) - L(f, \cP) \lt \epsilon.\]
  7. Let $\sT$ be the set of tagged partitions of $[a, b]$. For $(\cP_1, \cT_1), (\cP_2, \cT_2) \in \sT$, let us write $(\cP_1, \cT_1) \leq (\cP_2, \cT_2)$ if and only if $\cP_2$ is a strict refinement of $\cP_1$ or the tagged partitions are equal. Show that $\leq$ is a partial order on $\sT$, which makes it into a directed set.
  8. Show that $f$ is integrable with integral $I$ if and only if \[ \lim_{(\cP, \cT) \in \sT} R(f, \cP, \cT) = I. \]