

%This is how you make a comment.

\documentclass{article}

%-------Packages---------
\usepackage{amssymb,amsfonts,amsmath}
\usepackage{hyperref}
\usepackage[]{amsthm} %lets us use \begin{proof}

%--------Theorem Environments--------
%theoremstyle{plain} --- default
\newtheorem{thm}{Theorem}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{lem}[thm]{Lemma}

\newtheorem{quest}[thm]{Question}

\theoremstyle{definition}
\newtheorem{defn}[thm]{Definition}

\newtheorem{exer}{Exercise}



\bibliographystyle{plain}

%--------Meta Data: Fill in your info------
\title{Math 3100 Homework Assignment 5 - Due Friday March 12th}

\author{Your name here}


\date{\today}

\begin{document}

%\begin{abstract}

%This is where you would add an abstract

%\end{abstract}

\maketitle

%\tableofcontents


\begin{exer}
Suppose $f: [a, b] \to [a, b]$ is continuous. Prove that there exists $c \in [a, b]$ such that $f(c) = c$. 
\end{exer}


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





\begin{exer}
Prove that there does not exist a continuous function $f: \mathbb R \to \mathbb R$ such that the equation $f(x) = c$ has exactly two solutions for each $c \in \mathbb R$. 
\end{exer}





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


\begin{exer}
Find a function $f:[0, 1] \to \mathbb R$ that is not continuous, but still satisfies the intermediate value property, i.e., for each $0 \leq s < t \leq 1$ and $c \in [f(s), f(t)] \cup [f(t), f(s)]$, there exists a point $r \in [s, t]$ such that $f(r) = c$. 
\end{exer}




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

\begin{exer}
Show that if a function $f:[0, 1] \to \mathbb R$ satisfies the intermediate value property, then either $f$ is continuous or else there exists some $c \in \mathbb R$ such that the equation $f(x) = c$ has infinitely many solutions. 
\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}

