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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.