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\begin{document}
\begin{center}
{\bf \large Math 701 Problem Set 3} \vs \\
{\bf due Friday 9/27/2013} \vs \vs
\end{center}

\noindent
{\bf Problem 1.} Suppose $E$ and $F$ are closed sets in a metric space, $f:E \to Y$ and $g:F \to Y$ are continuous, and $f=g$ on $E \cap F$. Prove that the map $h: E \cup F \to Y$ defined by $h=f$ on $E$ and $h=g$ on $F$ is continuous. Show by an example that the assumption of $E,F$ being closed is essential. (Hint: Verify that for every closed set $C \subset Y$, $h^{-1}(C)$ is closed in $E \cup F$.) \vs 

\noindent
{\bf Problem 2.} Let $X$ be a metric space and $E \subset X$ be non-empty. Define the {\bit distance} between $x \in X$ and $E$ by
$$
\dist(x,E) = \inf_{p \in E} d(x,p).
$$
Show that $x \mapsto \dist(x,E)$ is uniformly continuous on $X$, and vanishes precisely on $\ov{E}$. \vs

\noindent
{\bf Problem 3.} Show that balls in $\RR^n$ (with the standard metric) are connected. Give an example of a metric space in which there are disconnected balls. \vs

\noindent
{\bf Problem 4.} Suppose $U$ is an {\it open} and connected set in $\RR^n$. Prove that $U$ is path-connected. (Hint: Fix a base point $p_0 \in U$ and let $E$ be the set of all points in $U$ that can be joined to $p_0$ by a path in $U$. Show that $E$ is both open and closed in $U$.)  \vs 

\noindent
{\bf Problem 5.} 
\begin{enumerate}
\item[(i)]
Let $f$ be a continuous map from a connected metric space $X$ to a metric space $Y$. Show that the {\bit graph} of $f$ defined by  
$$
\Gamma(f)= \{ (x,y) : y=f(x) \} \subset X \times Y
$$
is connected. Here on $X \times Y$ you can put any of the several equivalent metrics such as $d((x_1,y_1),(x_2,y_2))=d_X(x_1,x_2)+d_Y(y_1,y_2)$. \vs
\item[(ii)]
Show that the subset 
$$
E = \big\{ (x,y) \in \RR^2 : x > 0 \ \text{and} \ y = \sin (1/x) \big\} \cup \big\{ (0,y): -1 \leq y \leq 1 \big\} 
$$
of the plane is connected but not path-connected. 
\end{enumerate}
(Hint: For (i), note that $\Gamma(f)$ is the image of $X$ under the map $x \mapsto (x,f(x))$. For (ii), observe that $E$ is the closure of a graph.) \vs

\noindent
{\bf Problem 6.} 
\begin{enumerate}
\item[(i)]
Suppose $f: \RR \to \RR$ is differentiable and $\sup_{x \in \RR} |f'(x)|<+\infty$. Show that $f$ is uniformly continuous on $\RR$. \vs
\item[(ii)]
Suppose $f: \RR \to \RR$ is continuous, monotonic, and bounded. Show that $f$ is uniformly continuous on $\RR$. \vs
\item[(iii)]
Give an example of a bounded continuous function $f:\RR \to \RR$ which is not uniformly continuous. \vs
\end{enumerate}  

\end{document}

    
