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\title{MTH436 - Homework 2}

\begin{document}
 \maketitle
 
 Solutions to the questions in Section B should be submitted by the start of class on 2/13/19.
 
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 \section{Warm-up Questions}
 
 \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} Let $f \colon [0,1] \to \R$.
 \begin{enumerate}
  \item  Suppose that $f$ takes each value exactly twice. Show that $f$ is not a continuous function.
  \item  Suppose $f(0) < f(1)$ and for each $y \in \R$, the set $f^{-1}(y)$ consists of at most one element. Prove that $f$ is strictly increasing.
 \end{enumerate}
 \end{que}
 
 \begin{que}\mbox{}
 \begin{enumerate}
  \item Let $(X,d_X)$ be a metric space and $f,g \colon X \to \R$ be continuous.
  \begin{enumerate}
  \item Prove that $f+g$ is continuous.
  \item Prove that $fg$ is continuous.
  \end{enumerate}
   \item Let $(X,d_X)$, $(Y,d_Y)$ and $(Z,d_Z)$ be metric spaces and suppose $f \colon X \to Y$ and $g \colon Y \to Z$ be continuous. Prove that $g \circ f$ is continuous.
  \end{enumerate}
 \end{que}

 \begin{que}
  Find all continuous functions $f \colon (\R,|\cdot|) \to (\R,\delta)$ where $\delta$ is the discrete metric. Then find all continuous functions $f \colon (\R,\delta) \to (\R,|\cdot|)$.
 \end{que}

  \begin{que}
  Let $A$ be a subset of the metric space $X$.
  \begin{enumerate}
   \item Prove that $x \in \cl A$ if and only if for all $\eps > 0$ we have $V_\eps(x) \cap A \neq \varnothing$.
   \item Prove that $x \in \cl A$ if and only if there exists a sequence in $A$ which converges to $x$.
   \item Prove that $x \in b(A)$ if and only if for all $\eps > 0$ we have $V_\eps(x) \cap A \neq \varnothing$ and $V_\eps(x) \cap (X - A) \neq \varnothing$.
  \end{enumerate}

 \end{que}
 
 \begin{que}
  Let $(X,d)$ be a metric space. Define the distance between $x \in X$ and $A \subseteq X$ as $\dist(x,A) = \inf \{d(x,a) \mid a \in A \}$. Prove that if $A \subseteq X$ then $\cl A = \{ x \in X \mid \dist(x,A) = 0\}$.

 \end{que}




 
 
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 \section{Submitted Questions}
 
 \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}
 
 \begin{que} Let $(X,d)$ be a metric space.
  \begin{enumerate}
   \item Prove that $|\dist(x,A) - \dist(y,A)| \leq d(x,y)$ for all $x,y \in X$.
   \item Prove that $f \colon X \to \R$, $f(x) = \dist(x,A)$ is continuous.
   \end{enumerate}
   Now suppose $A,B$ are disjoint closed subsets of $X$. Show that there exists a continuous function $f \colon X \to \R$ satisfying
  \begin{enumerate}
   \item[(a)] $0 \leq f(x) \leq 1$ for all $x \in X$.
   \item[(b)] $f(a) = 0$ for all $a \in A$.
   \item[(c)] $f(b) = 1$ for all $b \in B$.
  \end{enumerate}
(Hint: Try $f(x) = \frac{\dist(x,A)}{\dist(x,A) + \dist(x,B)}$ and check it satisfies the desired properties.)
   \end{que}
 
 
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 \section{Challenge Questions}
 
 \begin{que}
  Suppose $(X,d)$ is a complete metric space and $Y \subseteq X$. Show that $(Y,d)$ is complete if and only if $Y$ is closed in $X$.
 \end{que}

 
 \begin{que}
  Let $f \colon (X,d_X) \to (Y,d_Y)$ be a continuous function and $A$ a dense subset of $X$.
  \begin{enumerate}
   \item Show that if $f$ is surjctive then $f(A)$ is dense in $Y$.
   \item Show that if $f$ is constant on $A$, then $f$ is constant on $X$.
  \end{enumerate}

 \end{que}

 
 \end{document}
