% HMC Math dept HW template example
% v0.04 by Eric J. Malm, 10 Mar 2005
\documentclass[12pt,letterpaper,boxed]{hmcpset}

% set 1-inch margins in the document
\usepackage[margin=1in]{geometry}

% include this if you want to import graphics files with /includegraphics
\usepackage{graphicx}

% include this to have lists enumerated by letters
\renewcommand{\labelenumi}{(\alph{enumi})}

% info for header block in upper right hand corner
\name{}
\class{Math 315}
\assignment{Homework \#14}
\duedate{5/30/2017}

% macros
\newcommand{\NN}{\mathbb{N}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\II}{\mathbb{I}}

\begin{document}

\problemlist{Abbott: 4.5.3, 4.5.7, 5.2.7, 5.2.10}


\begin{problem}[4.5.3]
A function $f$ is \emph{increasing} on $A$ if $f(x) \leq f(y)$ for all $x < y$ in $A$. Show that if $f$ is increasing on $[a,b]$ and satisfies the intermediate value property (Definition 4.5.3), then $f$ is continuous on $[a,b]$.
\end{problem}

%\begin{solution}
%
%\end{solution}

\newpage

\begin{problem}[4.5.7]
Let $f$ be a continuous function on the closed interval $[0,1]$ with range also contained in $[0,1]$. Prove that $f$ must have a fixed point; that is, show $f(x) = x$ for at least one value of $x \in [0,1]$.
\end{problem}

%\begin{solution}
%
%\end{solution}

\newpage

\begin{problem}[5.2.7]
Let
\[ g_a(x) = \begin{cases}
x^a \sin(1/x) & \text{if $x \neq 0$} \\
0 & \text{if $x = 0$.}
\end{cases} \]
Find a particular (potentially noninteger) value for $a$ so that
\begin{enumerate}
\item $g_a$ is differentiable on $\RR$ but such that $g'_a$ is unbounded on $[0,1]$.
\item $g_a$ is differentiable on $\RR$ with $g'_a$ continuous but not differentiable at zero.
\item $g_a$ is differentiable on $\RR$ and $g'_a$ is differentiable on $\RR$, but such that $g''_a$ is not continuous at zero.
\end{enumerate}
\end{problem}

%\begin{solution}
%
%\end{solution}

\newpage

\begin{problem}[5.2.10]
Recall that a function $f \colon (a,b) \to \RR$ is \emph{increasing} on $(a,b)$ if $f(x) \leq f(y)$ whenever $x < y$ in $(a,b)$. A familiar mantra from calculus is that a differentiable function is increasing if its derivative is positive, but this statement requires some sharpening in order to be completely accurate.

Show that the function
\[ g(x) = \begin{cases}
x/2 + x^2 \sin(1/x) & \text{if $x \neq 0$} \\
0 & \text{if $x = 0$}
\end{cases} \]
is differentiable on $\RR$ and satisfies $g'(0) > 0$. Now, prove that $g$ is \emph{not} increasing over any open interval containing $0$.

In the next section we will see that $f$ is indeed increasing on $(a,b)$ if and only if $f'(x) \geq 0$ for all $x \in (a,b)$.
\end{problem}

%\begin{solution}
%
%\end{solution}


\end{document}
