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\begin{center}\textbf{Problem Set 6 \hspace{.7in} Real Analysis 1 \hspace{.7in} Due 10/14/24} \end{center}

\noindent Five of these problems will be graded, with each problem worth 4 points. Clear and complete justification is required for full credit. You are welcome to discuss these problems with anyone and everyone, but must write up your own final submission without reference to any sources other than the textbook and instructor.

\begin{enumerate}
\item If $E \subseteq \mathbb{R}$ is closed, then $\mathbb{R} \backslash E$ is open.

\item If $\mathbb{R} \backslash E$ is open, then $E \subseteq \mathbb{R}$ is closed.

\item Show that if $f$ is continuous at $c \in [a,b]$ then for any sequence $\{x_n\}$ (with $x_n \in E,  \forall n \in \mathbb{N}$) converging to $c$, the sequence
$\{f(x_n)\}$ converges to $f(c)$.

\item Give an example of a function $f$ which is continuous on a bounded set $D$, but not bounded on $D$.


\end{enumerate}
For the following, prove or give a counterexample:
\begin{enumerate}[resume]
\item If a function $f$ is bounded and continuous on $D$, then $f$ is uniformly continuous on $D$.

\item If a function $f$ is continuous at a real number $x=a$, then $[f(x)]^2$ is also continuous at $x=a$.

\item The set $\{1,2,3 \}$ is closed.

\item The set $\left \{ \frac{1}{n} | n \in \mathbb{N} \right \}$ is closed.

\item The set $(0, \infty)$ is open.

\item The union of any two open sets is open.

\end{enumerate}

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