\documentclass[letterpaper,12pt]{article}

\usepackage{amsfonts,amssymb,amsmath,amsthm,txfonts,pxfonts,mathrsfs,fullpage,enumitem}
\usepackage{xcolor}

\pagestyle{empty}
\begin{document} 

\begin{center}\textbf{Problem Set 5 \hspace{.7in} Real Analysis 1 \hspace{.7in} Due 10/12/20} \end{center}

\noindent Four of these problems will be graded, with each problem worth 5 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 The sum of continuous functions is continuous.

\item The product of continuous functions is continuous.

\item The function $f(x)=x$ is continuous.

\item Let $c \in \mathbb{R}$.  The function $g(x)=c$ is continuous. 

\item Suppose that $D$ is the domain of $f$.  If $f$ is continuous at $a$, then there exists $\delta>0$ such that $f$ is bounded on the set $(a-\delta,a+\delta)\cap D$.

\item Determine where the function \[f(x)=\begin{cases} \frac{\sin x}{x}&\mathrm{if } \, x \neq 0\\
\enspace 1&\mathrm{if } \, x=0 \end{cases}\]
is continuous.

\item Determine where the function \[f(x)=\begin{cases} x \sin \frac{1}{x}&\mathrm{if } \, x \neq 0\\
\enspace 0&\mathrm{if } \, x=0 \end{cases}\]
is continuous.

\item Determine where the function \[f(x)=\begin{cases} x^2 \sin \frac{1}{x}&\mathrm{if } \, x \neq 0\\
\enspace 0&\mathrm{if } \, x=0 \end{cases}\]
is continuous.

\item $fg$ and $f$ continuous on $(a,b)$ implies that $g$ is continuous on $(a,b)$.

\item $|f|$ continuous on $(a,b)$ implies that $f$ is continuous on $(a,b)$.

\end{enumerate}

\end{document}