Math 104 Assignment 8

1. Sequential characterization of continuity

   Prove that a function $f : X \to Y$ is continuous if and only if for every convergent sequence $(a_n)_n$ in $X$, the sequence $(f(a_n))_n$ is convergent in $Y$ with \[f\paren{\limni a_n} = \limni f(a_n).\]

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

3. Equivalent metrics

   Let $M$ be a set. Two metrics $d$ and $d'$ on $M$ are said to be strongly equivalent if there are constants $c, C \in \R_{\gt0}$ so that for every $x, y \in M$, \[cd(x, y) \leq d'(x, y) \leq Cd(x, y).\]

   Now, let $d$ and $d'$ be strongly equivalent metrics on $M$.

   1. Show that $U \subseteq M$ is open with respect to $d$ if and only if it is open with respect to $d'$.
   2. Suppose that $X$ is another set, with strongly equivalent metrics $d_X$ and $d_X'$. Show that $f : M \to X$ is continuous with respect to $d$ and $d_X$ if and only if it is continuous with respect to $d'$ and $d_X'$.

4. Continuous functions on dense sets

   Suppose that $f, g : X \to Y$ are continuous, and $E \subseteq X$ is dense.
   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.)

5. 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 $f$.)
   4. Show that if $f$ is merely assumed to be continuous, it may not have a continuous extension.

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.

• Coordinates functions

  1. Show that the metric $d_\infty$ on $\R^n$ defined by \[d_\infty\paren{\vec{x}, \vec{y}} = \max\set{\abs{x_i - y_i} \mid 1 \leq i \leq n}\] is strongly equivalent to the usual metric \[d\paren{\vec{x}, \vec{y}} = \paren{\sum_{i=1}^n \abs{x_i-y_i}^2}^{\frac12}.\]
  2. Suppose that $f : M \to \R^n$, and let $f_1, \ldots, f_n : M \to \R$ be the coordinates of $f$, so that \[f(x) = \paren{f_1(x), \ldots, f_n(x)}.\] Show that $f$ is continuous if and only if all the $f_i$ are.

• Distances

  Let $M$ be a metric space, and define a metric on $M\times M$ by \begin{align*} d_2 : (M \times M) \times (M \times M) &\longrightarrow \R_{\geq0}\\ ((x_1, y_1), (x_2, y_2)) &\longmapsto \sqrt{d_M(x_1, x_2)^2 + d_M(y_1, y_2)^2}. \end{align*} (You may take for granted that this is a metric; notice that if $M = \R$, $d_2$ is the usual distance on $\R^2$.)

  Prove that the original metric on $M$, $d_M : M \times M \to \R_{\geq0}$, is continuous on the metric space $(M\times M, d_2)$.