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\title{Math 3110: Homework 7}
%\author{Zach Norwood}
\date{Due \textcolor{navy}{Wednesday, 20 March 2019}}

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\begin{document}

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\noindent
You will be graded on both the accuracy and the clarity of your solutions.
One purpose of the homework is to give you an opportunity to practice your proof-writing skills.

You are welcome\,---\,encouraged, even!\,---\,to collaborate on homework, but you should not 
copy solutions from any source, nor should you submit anything that you don't understand.

\addtocounter {problem} {-1}

\begin {problem} [don't submit any of this for a grade]
 This problem consists of some ``definition-chasing'' exercises that should help you feel more
 comfortable with the new concepts.
 \begin{enumerate} [(a)]
 \addtolength{\itemsep}{2pt}
 \item
 Suppose that $f\colon A\to\R$ is a function and $B\subs A$. The \emph{restriction} of $f$ to $B$,
 denoted $f\restr B$, is the function with domain $B$ given by, for $x\in B$, 
 \[ (f\restr B) (x) = f(x) . \]
 (It's just ``the same function'' with a restricted domain.)
 Prove that if $f$ is continuous, then so is $f\restr B$.
 \item
 Verify that, in the definition of \emph{continuity}, if a $\delta$ ``response'' meets an $\eps$ ``challenge,''
 then any smaller $\delta'$ also meets the same $\eps$ challenge.
 Verify also that $\delta$ meets any challenge $\eps'>\eps$.
 \item
 Let $f\colon A\to \R$ be a function.
 Verify that the following definitions of continuity are equivalent.
 \begin{enumerate} [(i)]
 \addtolength{\itemsep}{2pt}
  \item for all $c\in A$, for all $\eps>0$, there is $\delta>0$ such that \\
  	\phantom{m} for all $x\in A$, $\abs{x - c} < \delta$ implies $\abs{ f(x) - f(c) } < \eps$.
 \item for all $c\in A$, for all $\eps\in(0,1)$, there is $\delta>0$ such that \\
  	\phantom{m} for all $x\in A$, $\abs{x - c} < \delta$ implies $\abs{ f(x) - f(c) } < \eps$.
  \item for all $c\in A$, for all $\eps>0$, there is $\delta>0$ such that \\
  	\phantom{m} for all $x\in A\cap B_\delta(c)$, we have $f(x) \in B_\eps(f(c))$.
  \item for all $c\in A$, for all $\eps>0$, there is $\delta>0$ such that \\
  	\phantom{m} $f[ B_\delta(c)\cap A ] \subs B_\eps(f(c))$.
  \item for all $c\in A$, for all $\eps>0$, there is $\delta>0$ such that \\
  	\phantom{m} for all $x,y\in A\cap B_\delta(c)$, we have $\abs{ f(x) - f(y) }<\eps$.
 \end {enumerate}
 \item
 Show that for a function $f\colon\R\to\R$, the following are equivalent to continuity.
 \begin{enumerate} [(i), resume]
  \addtolength{\itemsep}{2pt}
  \item for all $c\in \R$, for all $\eps>0$, there is $\delta>0$ such that \\
  	\phantom{m} for all $x$, $\abs{x - c} < \delta$ implies $\abs{ f(x) - f(c) } < \eps$.
  \item for all open sets $U\subs\R$, the preimage $f\inv[U]$ is open.
  \item for all sets $B\subs\R$, we have $f[\bar{B}] \subs \bar{f[B]}$
 \end{enumerate}
 \end{enumerate}
\end {problem}

\newpage

\begin {problem}
 Prove that if $f\colon A\to \R$ and $g\colon B\to\R$ are continuous, then so are the functions
 \begin{enumerate} [(a)]
  \item $\abs{f}$,
  \item $\max(f,g)$, and
  \item $\min(f,g)$.
 \end{enumerate}
 (One might deduce (b) \& (c) from (a)\dots)
\end {problem}

\begin {problem}
 In class, we proved that if $f\colon A\to\R$ and $g\colon B\to\R$ are each continuous at $c\in A\cap B$,
 then the pointwise sum
 $f+g\colon A\cap B\to \R$ is continuous at $c$. Similar facts are also true for products and quotients
 of functions. Prove them. (\emph{Hint}: Use the sequential characterization of continuity.)
\end {problem}

\begin{problem}
Give a careful proof that all polynomials and rational functions are continuous,
using what we proved in lecture. That is, assuming that constant functions and the function $f(x) = x$ are 
continuous and that sums, products, and quotients of continuous functions are continuous,
prove that all polynomials and all rational functions are continuous.
\end{problem}

\begin {problem}
 Give an alternative proof of the Intermediate Value Theorem using the Nested Intervals Theorem.
 
 (\emph{Hint}: Google ``bisection method.'' Also, there are hints in the textbook.)
\end {problem}

\begin{problem}
Prove that if $f\colon [a,b]\to [a,b]$ is continuous, then there is $x\in[a,b]$ satisfying the equation 
$f(x) = x$.
\end{problem}

\begin {problem}
 For each of the following sets $A$, construct a continuous function $f\colon A \to \R$ that is unbounded 
 on $A$. Explain why this does not contradict the Boundedness Theorem.
 \begin{enumerate} [(a)]
  \item $A = \Z$;
  \item $A = (0,1)$;
  \item $A = [1,2]\cap\Q$.
 \end {enumerate}
\end {problem}

\begin {problem}
Prove that if $f\colon [a,b]\to \R$ is a one-to-one continuous function, 
then its inverse $f\inv$ is a continuous function.

(\emph{Hint}: A nice special case to start with is the square-root function.
See if you can prove that that function is continuous. If you get stuck, it's done in the textbook.)
\end {problem}

\end{document}




































