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\begin{document}
\begin{center}
{\bf \large Math 701 Problem Set 4} \vs \\
{\bf due Friday 10/4/2013} \vs \vs
\end{center}

\noindent
{\bf Problem 1.} Suppose $-\infty<a<b<+\infty$ and $f: (a,b) \to \RR$ is uniformly continuous. Show that $\lim_{x \to a^+} f(x)$ and $\lim_{x \to b^-} f(x)$ exist. \vs

\noindent
{\bf Problem 2.} Let $X$ be a metric space. \vs
\begin{enumerate}
\item[(i)]
Suppose $E,F$ are disjoint closed subsets of $X$. Show that there are disjoint open sets $U,V$ in $X$ such that $E \subset U$ and $F \subset V$. \vs

\item[(ii)]
Suppose $p_1, \ldots, p_n$ are distinct points in $X$ and $a_1, \ldots, a_n$ are real numbers. Show that there exists a continuous function $f: X \to \RR$ such that $f(p_i)=a_i$ for every $1 \leq i \leq n$.  \vs 
\end{enumerate}
(Hint: Use Urysohn's lemma for both parts. In (ii), first argue that there exists a continuous function on $X$ which takes the value $1$ at $p_i$ and vanishes at $p_j$ for all $j \neq i$.) \vs

\noindent
{\bf Problem 3.} A subset of $\RR$ is called a {\bit $G_\delta$-set} if it is a countable intersection of open sets. \vs
\begin{enumerate}
\item[(i)]
Show that every closed set in $\RR$ is a $G_\delta$-set. \vs
\item[(ii)]
Show that $\RR \sm \QQ$, the set of irrational numbers, is a $G_\delta$-set. \vs 
\item[(iii)]
By contrast, show that $\QQ$ is not a $G_\delta$-set. \vs
\end{enumerate}
(Hint: For (i), consider $\ve$-neighborhoods. For (iii), use Baire's theorem.) \vs  

\noindent
{\bf Problem 4.} Given a function $f: \RR \to \RR$, let $C(f)$ denote the set of points at which $f$ is continuous. \vs

\begin{enumerate}
\item[(i)]
Prove that   
$$
C(f)= \bigcap_{n \geq 1} \big \{ x \in \RR : \diam f((x-r,x+r)) < \tfrac{1}{n} \ \text{for some} \ r>0 \big\}. 
$$
In particular, $C(f)$ is always a $G_\delta$-set. \vs

\item[(ii)]
Conclude that there is no $f:\RR \to \RR$ with $C(f)=\QQ$. \vs 
 
\item[(iii)]
By contrast, give an example $f: \RR \to \RR$ with $C(f)=\RR \sm \QQ$. \vs
\end{enumerate}

\noindent
{\bf Problem 5.} 
Suppose $f: \RR \to \RR$ is continuous and $\lim_{n \to \infty} f(nx)=0$ for every $x \in \RR$. Show that $\lim_{x \to \infty} f(x)=0$. (Hint: For a given $\ve>0$, apply Baire's theorem to the sets 
$$
F_k = \{ x \in \RR: |f(nx)| \leq \ve \ \text{for all} \ n > k \} \qquad k=1,2,3,\ldots) \vs
$$

\noindent
{\bf Problem 6.} Suppose $X$ is a complete metric space and ${\mathscr F}$ is a family of continuous functions $X \to \RR$. Assume $\mathscr F$ is a pointwise bounded family in the following sense: For each $x \in X$, there is a constant $M_x>0$ such that $|f(x)|\leq M_x$ for every $f \in {\mathscr F}$. Show that there is a non-empty open set $U \subset X$ and a number $M>0$ such that $|f(x)|\leq M$ for every $f \in {\mathscr F}$ and every $x \in U$. This is often known as the {\bit principle of uniform boundedness}. (Hint: Use Baire's theorem.) 




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