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\usepackage{amssymb,amsfonts,amsmath}
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\title{Math 3100 Homework Assignment 4 - Due Wednesday March 3rd}

\author{Your name here}


\date{\today}

\begin{document}

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\begin{exer}
Let $f: \mathbb R \to \mathbb R$ be 1-1 and locally monotone in the sense that for every point $t_0 \in \mathbb R$ there exists $\varepsilon > 0$ such that the restriction of $f$ to the interval $(t_0 - \varepsilon, t_0 + \varepsilon)$ is monotone. Prove that $f$ is monotone on all of $\mathbb R$. 
\end{exer}


\noindent\makebox[\linewidth]{\rule{\textwidth}{0.4pt}}





\begin{exer}
A function $f: \mathbb R \to \mathbb R$ is periodic if there exists some $t_0 > 0$ such that $f(x + t_0) = f(x)$ for all $x \in \mathbb R$. Prove that if $f: \mathbb R \to \mathbb R$ is periodic and continuous then $f$ is uniformly continuous. 
\end{exer}





\noindent\makebox[\linewidth]{\rule{\textwidth}{0.4pt}}

\begin{exer}
Suppose $A, B \subset \mathbb R$ are sets and $f: A \cup B \to \mathbb R$ is continuous such that $f_{| A}$ and $f_{|B}$ are each uniformly continuous. Must $f$ be uniformly continuous? Either prove or give a counter-example. What if $A$ and $B$ are closed subsets?
\end{exer}




\noindent\makebox[\linewidth]{\rule{\textwidth}{0.4pt}}

\begin{exer}
Let $\{ G_\alpha \}_{\alpha \in A}$ be an arbitrary family of open subsets of $\mathbb R$. Prove that $\cup_{\alpha \in A} E_\alpha$ is open. Also, show that if $G_1, \ldots, G_n \subset \mathbb R$ are open sets then $\cap_{i = 1}^n G_i$ is open. 
\end{exer}





\noindent\makebox[\linewidth]{\rule{\textwidth}{0.4pt}}

\begin{exer}
Let $A \subset \mathbb R$ be a bounded set and suppose $f: A \to \mathbb R$ and $g: A \to \mathbb R$ are each uniformly continuous. Show that $fg: A \to \mathbb R$ given by $(fg)(t) = f(t)g(t)$ is also uniformly continuous. 
\end{exer}




%\begin{thebibliography}{9}



%\bibitem{halmosmath}
%Paul R. Halmos
%How to write mathematics.
%\url{https://sites.math.washington.edu/~lind/Resources/Halmos.pdf}

%\bibitem{knuthmath}
%Donald E. Knuth, Tracy Larrabee and Paul M. Roberts
%Mathematical Writing.
%\url{https://jmlr.csail.mit.edu/reviewing-papers/knuth_mathematical_writing.pdf}

%\end{thebibliography}

\end{document}

