$$ \newcommand{\cis}{\operatorname{cis}} \newcommand{\norm}[1]{\left\|#1\right\|} \newcommand{\paren}[1]{\left(#1\right)} \newcommand{\sq}[1]{\left[#1\right]} \newcommand{\abs}[1]{\left\lvert#1\right\rvert} \newcommand{\set}[1]{\left\{#1\right\}} \newcommand{\ang}[1]{\left\langle#1\right\rangle} \newcommand{\floor}[1]{\left\lfloor#1\right\rfloor} \newcommand{\ceil}[1]{\left\lceil#1\right\rceil} \newcommand{\C}{\mathbb{C}} \newcommand{\D}{\mathbb{D}} \newcommand{\R}{\mathbb{R}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\N}{\mathbb{N}} \newcommand{\F}{\mathbb{F}} \newcommand{\T}{\mathbb{T}} \renewcommand{\S}{\mathbb{S}} \newcommand{\intr}{{\large\circ}} \newcommand{\limni}[1][n]{\lim_{#1\to\infty}} \renewcommand{\Re}{\operatorname{Re}} \renewcommand{\Im}{\operatorname{Im}} \newcommand{\cA}{\mathcal{A}} \newcommand{\cB}{\mathcal{B}} \newcommand{\cC}{\mathcal{C}} \newcommand{\cD}{\mathcal{D}} \newcommand{\cE}{\mathcal{E}} \newcommand{\cF}{\mathcal{F}} \newcommand{\cG}{\mathcal{G}} \newcommand{\cH}{\mathcal{H}} \newcommand{\cI}{\mathcal{I}} \newcommand{\cJ}{\mathcal{J}} \newcommand{\cK}{\mathcal{K}} \newcommand{\cL}{\mathcal{L}} \newcommand{\cM}{\mathcal{M}} \newcommand{\cN}{\mathcal{N}} \newcommand{\cO}{\mathcal{O}} \newcommand{\cP}{\mathcal{P}} \newcommand{\cQ}{\mathcal{Q}} \newcommand{\cR}{\mathcal{R}} \newcommand{\cS}{\mathcal{S}} \newcommand{\cT}{\mathcal{T}} \newcommand{\cU}{\mathcal{U}} \newcommand{\cV}{\mathcal{V}} \newcommand{\cW}{\mathcal{W}} \newcommand{\cX}{\mathcal{X}} \newcommand{\cY}{\mathcal{Y}} \newcommand{\cZ}{\mathcal{Z}} \newcommand{\bA}{\mathbb{A}} \newcommand{\bB}{\mathbb{B}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bD}{\mathbb{D}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bG}{\mathbb{G}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bI}{\mathbb{I}} \newcommand{\bJ}{\mathbb{J}} \newcommand{\bK}{\mathbb{K}} \newcommand{\bL}{\mathbb{L}} \newcommand{\bM}{\mathbb{M}} \newcommand{\bN}{\mathbb{N}} \newcommand{\bO}{\mathbb{O}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bS}{\mathbb{S}} \newcommand{\bT}{\mathbb{T}} \newcommand{\bU}{\mathbb{U}} \newcommand{\bV}{\mathbb{V}} \newcommand{\bW}{\mathbb{W}} \newcommand{\bX}{\mathbb{X}} \newcommand{\bY}{\mathbb{Y}} \newcommand{\bZ}{\mathbb{Z}} $$

Useful links

Office hours:

  • Mondays 9:00-10:00
  • Wednesdays 14:00-16:00

Email

GSI:

Rahul Dalal
  • M 10:30-12:30
  • TTh 17:30-19:30
  • WF 11:00-13:00

Exams

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.