Ian Charlesworth

Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces.

1. Suppose that $E \subseteq X$ and $x \in E \setminus E'$. Let $f : E \to Y$. The statement "$f$ is continuous at $x$" is...

   1. ...always true.
   2. ...nonsense since it only makes sense for $x \in E'$.
   3. ...nonsense since it only makes sense for $x \in E' \cap E$.
   4. ...potentially true or false.

2. Suppose that $g : \Z \to \R$ is given by \[g(z) = \begin{cases} 0 & \text{ if } z = 0 \\ \frac1{z^2} & \text{ otherwise.} \end{cases}\] Then $g$ is continuous...

   1. ...everywhere.
   2. ...everywhere except $0$.
   3. ...nowhere.
   4. Mu. $g$ is not the type of object for which continuity makes sense, so the question is malformed.

3. If $X$ is bounded and $g : X \to Y$ is continuous, then $g(X)$ is bounded.

   1. True.
   2. True, provided that $g$ is uniformly continuous.
   3. False.

4. Let $f : X \to Y$. Suppose $(a_n)_n$, $(b_n)_n$ are sequences in $\R_{\gt0}$ so that $(a_n)_n$ converges to $0$ and if $s, t \in X$ with $d_X(s, t) \lt b_n$ it follows that $d_Y( f(s), f(t) ) \lt a_n$. Which is the strongest true statement below?

   1. If $(b_n)_n$ also converges to zero, then $f$ is continuous.
   2. If $(b_n)_n$ also converges to zero, then $f$ is uniformly continuous.
   3. $f$ is continuous.
   4. $f$ is uniformly continuous.