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\title{MTH436 - Homework 1}

\begin{document}
 \maketitle
 
 Solutions to the questions in Section B should be submitted by the start of class on 2/7/18.
 
 
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 \section{Warm-up Questions}
 
 \begin{que}
  Decide if the following functions are uniformly continuous.
  \begin{enumerate}
   \item[(i)] $f(x) = \frac{1}{x^2}$ on $[2,\infty)$.
   \item[(ii)] $f(x) = \sin \left( \frac{1}{x} \right)$ on $(0,1)$.
   \item[(iii)] $f(x) = x^2$ on $\N$.
  \end{enumerate}

 \end{que}

 \begin{que}
  Given an example of a function $f \colon [a,b] \cap \Q \to \R$ which is continuous and bounded but not uniformly continuous.
 \end{que}

 \begin{que}
  Suppose $f, g \colon A \to \R$ are Lipschitz functions.
  \begin{enumerate}
   \item[(i)] Prove that $f + g$ is Lipschitz.
   \item[(ii)] Prove that if $f$ and $g$ are bounded functions then $fg$ is Lipschitz.
   \item[(iii)] Give an example to show that $fg$ is not necessarily Lipschitz.
  \end{enumerate}

 \end{que}

 
 \begin{que}
  Suppose $A \subseteq \R$ is bounded and $f \colon A \to \R$ is uniformly continuous. Prove that $f$ is bounded.
 \end{que}

 \begin{que}
  Suppose $I$ is an interval and $f \colon I \to \R$ is increasing. For $c \in I$, prove the following.
  \begin{enumerate}
   \item[(i)] $j_f(c) \geq 0$.
   \item[(ii)] $f$ is continuous at $c$ if and only if $j_f(c) = 0$.
  \end{enumerate}
 \end{que}

 
 \begin{que}
  Suppose $f \colon [a,b] \to \R$ is continuous and attains its absolute maximum at $c \in (a,b)$. Show that $f$ is not injective.
 \end{que}

 
 
 
 
 
 
 
 
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 \section{Submitted Questions}
 
 
 \begin{que}
  Let $I$ be an interval. We say that $f \colon I \to \R$ is \emph{absolutely continuous} if for all $\eps > 0$ there exists $\delta > 0$ such that if $\{ [x_k,y_k] \mid k =1,\dotsc,n\}$ is a finite collection of pairwise disjoint subintervals with $\sum_{k=1}^n |x_k - y_k| < \delta$ then $\sum_{k=1}^n |f(x_k) - f(y_k)| < \eps$.
  \begin{enumerate}
   \item[(i)] Prove that if $f$ is absolutely continuous, it is uniformly continuous.
   \item[(ii)] Prove that if $f$ is Lipschitz, then it is absolutely continuous.
  \end{enumerate}

 \end{que}

\begin{que} Let $f \colon [0,1] \to \R$. 
  \begin{enumerate}
   \item[(i)] Suppose that $f$ takes each value exactly twice. Show that $f$ is not a continuous function.
   \item[(ii)] Suppose $f(0) < f(1)$ and for each $y \in \R$, the set $f^{-1}(y)$ consists of at most one element. Prove that if $f$ is continuous then $f$ is strictly increasing.
  \end{enumerate}

\end{que}
 
 
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 \section{Challenge Questions}
 
 \begin{que}
  Let $f \colon [-1,1] \to \R$ be defined by
  \[
   f(x) = \begin{cases}
	      x \sin \left( \frac{1}{x} \right) \quad &\text{if $x \neq 0$}, \\
	      0 \quad &\text{if $x=0$}.
	  \end{cases}
  \]
Prove that $f$ is uniformly continuous but not absolutely continuous.
 \end{que}

\begin{que}
 \emph{The Cantor set}. We will construct a (relatively pathological) closed set in $\R$. We define the \emph{Middle-thirds Cantor set} to be
 \[
  C = \left\{ x \in [0,1] \middle| x = \sum_{n=1}^{\infty} \frac{a_n}{3^n}, \text{ where $a_n \in {0,2}$ for all $n$}\right\}.
 \]
We can construct $C$ using an iterative method as follows. Let $C_0 = [0,1]$, and from $C_0$ remove the middle-third interval $(1/3,2/3)$ to obtain $C_1 = [0,1/3] \cup [2/3,1]$. From each of the two intervals making up $C_1$, remove the middle third interval to give the set $C_2 = [0,1/9] \cup [2/9,1/3] \cup [2/3,7/9] \cup [8/9,1]$. Continue in this way, removing the middle third of each of the $2^n$ intervals of $C_n$ at each step. The eventual set you are left with is in fact $C$: formally $C = \cap_{i=1}^{\infty} C_i$.
\begin{itemize}
 \item[(i)] Draw the sets $C_n$ for $n=1,2,3,4$.
 \item[(ii)] Prove that $C$ is closed.
 \item[(iii)] What is the sum of the length of all the removed intervals? What does this suggest about the ``length'' of $C$?
 \item[(iv)] Prove that $C$ contains no open interval.
 \item[(v)] Show that $\phi \colon C \to [0,1]$ given by
 \[
  \phi \left( \sum_{n=1}^{\infty} \frac{a_n}{3^n} \right) = \sum_{n=1}^{\infty} \frac{a_n}{2^n}
 \]
is a surjection, and deduce that $C$ is uncountable. Does this surprise you with regards to your conclusion in part (iii)?
\end{itemize}

\end{que}

 
 
 
 
 \end{document}
 
 
 