$$ \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 18:00-19:00
  • Wednesdays 14:00-16:00

Email

GSI:

Nima Moini
  • Mondays 10:00-14:00
  • Tuesdays 10:00-14:00
  • Wednesdays 10:00-12:00

Exams

  1. Suppose that $(x_n)_n$ is a sequence in a metric space $(M, d)$. Which of the following is the condition that $(x_n)_n$ does not converge?
    1. There is some $L \in M$ and some $\delta \gt 0$ so that for any $N$ there is $n \gt N$ with $d(x_n, L) \gt \delta$.
    2. If $U \subseteq M$ is an open set, for any $N$ there is some $n \gt N$ so that $x_n \notin U$.
    3. For any $L \in M$ and any $\epsilon \gt 0$ there is some $N$ so that for every $n \gt N$, $d(x_n, L) \gt \epsilon$.
    4. For any $L \in M$ there is some $\delta \gt 0$ so that for any $N$ there is $n \gt N$ with $d(x_n, L) \gt \delta$.
  2. Let $X$ be a set equipped with the discrete metric. Suppose $f : X \to \R$. What condition is necessary for $f$ to be continuous?
    1. $f$ must be constant.
    2. No further condition is needed.
    3. We must have $\lim_{t \to x} f(t) = f(x)$ for every $x \in X$.
    4. $f$ must be bounded, in the sense that $f(X) = \set{f(t) \mid t \in X}$ is a bounded subset of $\R$.
  3. Let $X = \R\setminus\set0$ with the metric inherited from $\R$. Define $f : X \to \R$ by $f(t) = \frac{t}{\abs{t}}$. Is $f$ continuous?
    1. Yes.
    2. No.
  4. Let $g : \R\to\R$ be defined by $$g(t) = \begin{cases} 1 &\text{ if } t \in \Q \\ 0 &\text{ otherwise.}\end{cases}$$ Then $g$ is continuous...
    1. ...on $\Q$ but not on $\R\setminus\Q$.
    2. ...everywhere.
    3. ...nowhere.
    4. ...at $0$, but nowhere else.
  5. Let $h : \R\to\R$ be defined by $$h(t) = \begin{cases} t &\text{ if } t \in \Q \\ 0 &\text{ otherwise.}\end{cases}$$ Then $h$ is continuous...
    1. ...on $\R\setminus\Q$ but not on $\Q$.
    2. ...everywhere.
    3. ...nowhere.
    4. ...at $0$, but nowhere else.