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\begin{document}

\title{Homework \#7, Real Analysis I, Fall 2012}


\author{Marius Ionescu}
\date{\bf{Due Friday, 11/9/2012}}
\maketitle
\begin{problem}[Exercise 4.3.5 on page 113]
In Theorem 4.3.4, statement (iv) says that $f(x)/g(x)$ is continuous
at $c$ if both $f$ and $g$ are, provided that the quotient is
defined. Show that if $g$ is continuous at $c$ and $g(c)\ne 0$, then
there exists an open interval containing $c$ on which $f(x)/g(x)$ is
defined.
\end{problem}

\begin{problem}[Exercise 4.3.7 on page 113]
Assume that $h:\R\to\R$ is continuous on $\R$ and let
$K:=\{x\,:\,h(x)=0\}$. Show that $K$ is a closed set.
\end{problem}
%\begin{proof}
  
%\end{proof}
\begin{problem}[Exercise 4.4.4 on page 119]
  Show that if $f$ is continuous on $[a,b]$ with 
$f(x)>0$ for all $a\leq x\leq b$, then $1/f$ is bounded on 
$[a,b]$.
\end{problem}
\begin{problem}[Exercise 4.4.8 on page 119] 
\begin{itemize}
\item[(a)]  Assume that $f:[0,\infty)\to\R$ is continuous at 
every point in its domain.  Show that if there exists $b>0$ such 
that $f$ is uniformly continuous on the set $[b,\infty)$, then $f$ 
is uniformly continuous on $[0,\infty)$.
\item[(b)] Prove that $f(x) = \sqrt{x}$ is uniformly
continuous on $[0,\infty)$.
\end{itemize}
\end{problem}
\begin{problem}[Exercise 4.4.9 on page 120]
  A function $f:A\to\R$ is called \textit{Lipschitz}
if there exists a bound $M>0$ such that 
\[
\left|\frac{f(x)-f(y)}{x-y}\right| \leq M
\]
for all $x,y\in A$.  Geometrically speaking, a function $f$ is 
Lipschitz if there is a uniform bound on the magnitude of the 
slopes of lines drawn through any two points on the graph of $f$.
\begin{itemize}
\item[(a)] Show that if $f:A\to\R$ is Lipschitz, then it is
  uniformly continuous on $A$.
\item[(b)]  Is the converse statement true?  Are all uniformly 
  continuous functions necessarily Lipschitz?
\end{itemize}
\end{problem}
\begin{problem}[Exercise 4.5.3 on page 124]
  Is there a continuous function on all of
$\R$ with range $f(\R)$ equal to $\Q$?
\end{problem}
\end{document}
