Math 104 Assignment 8

1. Limits of functions

   1. Suppose $(M, d)$ is a metric space, $E \subseteq M$, $f, g : E \to \R$, and $p$ is a cluster limit point of $E$. Suppose further that for some $r \gt 0$, the set $f(E\cap B_r(p)) = \set{f(x) \mid x \in E\cap B_r(p)} \subseteq \R$ is bounded, and that \[\lim_{x\to p}g(x) = 0.\] Show that \[\lim_{x\to p} f(x)g(x) = 0.\] (Note: you should not assume that $f$ has a limit at $p$, unless you prove it from these hypotheses.)

   Suppose that $(M, d)$ and $(X, d_X)$ are metric spaces, $E \subseteq M$, $f : E \to X$, and $p$ is a cluster limit point of $E$.

   2. Show that if $f$ has a limit at $p$, then for every $\epsilon \gt 0$ there is some $r \gt 0$ so that whenever $x, y \in B_r(p) \cap E \setminus \set{p}$, $d_X(f(x), f(y)) \lt \epsilon$.
   3. Suppose that for every $\epsilon \gt 0$ there is some $r \gt 0$ so that whenever $x, y \in B_r(p) \cap E$, $d_X(f(x), f(y)) \lt \epsilon$. Show that if $X$ is complete, then $f$ has a limit at $p$.
   4. Show that part (C) may fail if $X$ is not assumed to be complete.

2. Composition of functions continuous at a point

   Suppose that $X, Y, Z$ are metric spaces, that $f : X \to Y$ and $g : Y\to Z$, that $x_0 \in X$, and that $f$ is continuous at $x_0$ while $g$ is continuous at $f(x_0)$. Verify that $g\circ f$ is continuous at $x_0$.

   (You should not assume that $f$ or $g$ are continuous anywhere else. If you use the open set chracterisation mentioned in extra problems below, you should prove it, although I don't think that is the easiest way forward for this problem.)

3. Continuity in higher dimensions

   Within this question, we will equip $\R^n$ with the Euclidean distance $d_2$ given by \[d_2(\vec x, \vec y) = \sqrt{\sum_{i=1}^n \abs{x_i-y_i}^2}.\]

   1. For each $i = 1, \ldots, n$, define \begin{align*} \pi_i : \R^n &\longrightarrow \R \\ (x_1, \ldots, x_n) &\longmapsto x_i. \end{align*} Prove that $\pi_i$ is continuous.
   2. Let $p : \R^n \to \R$ be a polynomial function (in $n$ variables). Prove that $p$ is continuous.

   3. Let $f : \R \to \R^n$ and for each $i$ define $f_i = \pi_i \circ f : \R \to \R$. Prove that if each $f_i$ is continuous, then so is $f$.

      (In fact, $f$ is continuous if and only if each $f_i$ is; the other direction follows immediately from part A and a statement we will prove soon, that the composition of continuous functions is continuous.)

   4. Define \begin{align*} g : \R^2 &\longrightarrow \R \\ (x, y) &\longmapsto \begin{cases} \frac{xy}{x^2+y^2} & \text{ if } x^2+y^2 \neq 0 \\ 0 & \text{ otherwise.}\end{cases} \end{align*} Show that for any fixed $x_0, y_0 \in \R$, the functions $x \mapsto g(x, y_0)$ and $y \mapsto g(x_0, y)$ are continuous, but $g$ is not. Show that $g$ is not continuous, but that for any fixed $x_0, y_0 \in \R$ the functions $x \mapsto g(x, y_0)$ and $y \mapsto g(x_0, y)$ are.

4. 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 continuous at every irrational number, but discontinuous at every rational number.

5. Continuous functions on dense sets

   Suppose that $f, g : X \to Y$ are continuous, and $E \subseteq X$ is dense; recall that by definition this means $\overline E = X$.

   (This time you may use the additional characterisations of density given below, but if you do you should convince yourself that you can prove them.)

   1. Prove that $f(E)$ is dense in $f(X)$.
   2. Prove that if $f(x) = g(x)$ for all $x \in E$, then $f(x) = g(x)$ for all $x \in X$. (Thus the value of a continuous function is determined by its values on a dense set.)

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.

• A characterisation of density

  Suppose that $(M, d)$ is a metric space and $X \subseteq M$. Prove that the following are equivalent:

  1. $X$ is dense in $M$, i.e., $M = \overline{X}$ (the closure of $X$);
  2. for every $a \in M$ and every $\epsilon \gt 0$ there is some $x \in X \cap B_\epsilon(a)$; and
  3. for every $a \in M$ there is some sequence in $X$ converging to $a$.
  (Warning: this is not the same as saying that every $a \in M$ is a cluster limit point of $X$; convince yourself that these are different.)

• Closures and limits

  I'm not sure if I proved these useful facts in lecture or not. Prove them now, without looking things up in the notes or the text.

  Suppose that $(M, d)$ is a metric space.

  1. Suppose that $F \subseteq M$ is closed, and $(x_n)_n$ is a sequence in $F$. Then if $(x_{n_k})_k$ is a convergent subsequence of $(x_n)_n$, its limit is in $F$.
  2. Suppose that $E \subseteq M$. Then $x \in \overline{E}$ if and only if there is a sequence in $E$ which converges to $x$.

• An open set characterisation of continuity at a point

  Suppose that $X, Y$ are metric spaces, $f : X \to Y$, and $x_0 \in X$. Show that $f$ is continuous at $x_0$ if and only if whenever $U \subseteq Y$ is open with $f(x_0) \in Y$, there is an open set $V \subseteq X$ with $x \in V \subseteq f^{-1}(U)$.

  This is unfortunately not as nice as the case for functions that are continuous on all of $X$.

• Distances, closed sets, and compact sets

  Suppose that $K \subseteq M$ is compact and $F \subseteq M$ is closed and bounded. Prove that there are $x \in K, y \in F$ so that \[d(x, y) = \inf\set{d(s, t) \mid s \in J, t \in K},\] or show by example that this may not be the case.

• Continuity and relatively open sets

  Let $(M, d)$ be a metric space, and $X \subseteq M$. Define $\iota$ to be the inclusion $\iota : X \hookrightarrow M$ defined by $x \mapsto x$. Ponder the relation between Assignment 4, Problem 3 and the continuity of $\iota$.

• Separability

  Recall that a set $S$ is said to be countable if there is an injective function $S \hookrightarrow \N$. A metric space $(M, d)$ is said to be separable if it has a countable dense subset.

  1. Prove that $\R$ is separable.
  2. Prove that $\R^n$ is separable for every $n$.
  3. Prove that every totally bounded metric space (hence every compact metric space) is separable.
  4. A metric space is said to be $\sigma$-compact if it is the countable union of compact sets. Prove that every $\sigma$-compact metric space is separable.
  5. Prove that $\R$ is $\sigma$-compact.
  6. Suppose that $(M, d)$ is a separable metric space. Prove that any collection of pairwise disjoint open subsets of $M$ is countable.
  7. A metric space $(M, d)$ is said to be second-countable if there is a countable collection $\mathscr{U}$ of open subsets of $M$ so that every open set $G \subseteq M$ can be written as the union of sets contained in $\mathscr{U}$. Prove that $M$ is second-countable if and only if it is separable.
  8. Let $\ell^\infty(\N) = \set{(a_n)_n \mid (a_n)_n \text{ is a bounded sequence in } \C}$, and define a metric on $\ell^\infty(\N)$ by \[d_\infty\paren{ (a_n)_n, (b_n)_n } = \sup_{n\in \N} \abs{a_n-b_n}.\] Then $(\ell^\infty(\N), d_\infty)$ is a metric space. Prove that it is not separable.