Math 104 Assignment 8

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

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. 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.)

4. Uniformly continuous functions have extensions

   Again, suppose that $E \subseteq X$ is dense, and suppose that $f : E \to Y$ is uniformly continuous on $E$. Suppose also that $Y$ is complete.

   1. Show that if $(x_n)_n$ is a Cauchy sequence in $E$, then $(f(x_n))_n$ is Cauchy in $Y$.
   2. Show that if $(x_n)_n$ and $(a_n)_n$ are sequences in $E$ converging to $x \in X$, then $(f(x_n))_n$ and $(f(a_k))_k$ converge in $Y$ to the same limit.
   3. Prove that there is a uniformly continuous function $\tilde f : X \to Y$ so that $\tilde f(x) = f(x)$ for all $x \in E$. (This function is called a continuous extension of $f$; by the previous problem, we know it is unique.)
   4. Show that if $f$ is merely assumed to be continuous, it may not have a continuous extension.

5. Distances and compact sets

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

   1. Let $d_2$ be the metric on $M \times M$ given by \[d_2( (x_1, x_2), (y_1, y_2) )^2 = d(x_1, y_1)^2 + d(x_2, y_2)^2.\] Prove that \[d : M\times M \to \R_{\geq0}\] is continuous with respect to $d_2$.
   2. Suppose that $J, K \subseteq M$ are compact and non-empty. Prove that there are $x \in J$, $y \in K$ so that \[d(x, y) = \inf\set{d(s, t) \mid s \in J, t \in K}.\] (Any two compact sets have a pair of points which are as close as possible.)

6. Meditations on a theme of discontinuity

   1. Let $S : \R \to \R$ be a continuous function with the property that for any $T \in \R$ there are $x, y \gt T$ and $a, b \lt T$ so that \[S(x) = 1 = S(a) \hspace{2cm}\text{and}\hspace{2cm} S(y) = -1 = S(b).\] Let $f : \R \to \R$ satisfy $f(x) = S\paren{\frac1x}$ for $x \neq 0$, and $f(0) = 0$. Determine where $f$ is continuous.
   2. Let $f$ be the function from Part A. Find an open set $U \subseteq \R$ so that $f^{-1}(U)$ is not open.
   3. Let $g : \R \to \R$ be given by \[g(x) = \begin{cases} 0 & \text{ if } x \lt 0 \\ 1 & \text{ otherwise}.\end{cases}\] Determine which open subsets of $\R$ have open preimages under $g$.
   

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 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 $\iota$.