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\pagestyle{fancy} \lhead{{\sf Homework 10 $|$ Due June 18 (Monday) }}
\rhead{\thepage} \cfoot{{\sf MthSc 453 $|$ Real Analysis I $|$ Summer
    I 2012 $|$ M.~Macauley}}


\begin{document}

$\;$ 

\noindent\emph{Read}: Rudin, Chapter 4. \\

\begin{enumerate}
  

\item Let $f$ and $g$ be continuous mappings of a metric space $X$
  into a metric space $Y$, and let $E$ be a dense subset of $X$. 
  \begin{enumerate}
  \item Prove that $f(E)$ is dense in $f(X)$. 
  \item If $g(p)=f(p)$ for all
    $p\in E$, prove that $g(p)=f(p)$ for all $p\in X$. (In other
    words, a continuous mapping is determined by its values on a dense
    subset of its domain.) \\
  \end{enumerate}
    


\item Let $f\colon X\to Y$ and $g\colon Y\to Z$ be uniformly
  continuous between metric spaces.
  \begin{enumerate}
  \item Prove that if $\{x_n\}$ is a Cauchy sequence in $X$, then
    $\{f(x_n)\}$ is Cauchy in $Y$.
  \item Prove that $g\circ f$ is uniformly continuous. \\
  \end{enumerate}



\item Prove the following fixed point theorem: If
  $f\colon[0,1]\to[0,1]$ is continuous, then $f(x)=x$ for at least one
  $x\in[0,1]$. [\emph{Hint}: Consider the function $g(x):=f(x)-x$.] \\



\item Every $x\in\Q$ can be written as $x=m/n$, where $n>0$, and $m$
  and $n$ are integers with no common divisors. When $x=0$, we take
  $n=1$. Consider the real-valued function $f$ defined by
  \[
  f(x)=\left\{\begin{array}{lll}0 && x\not\in\Q, \\ 1/n && x=m/n.
  \end{array}\right.
  \]
  Prove that $f$ is continuous at every irrational point, and that
  $f$ has a \emph{simple discontinuity} at every rational
  point. 
  
\end{enumerate}

\end{document}
