\documentclass[11pt]{article}
%Options: draft shows overfull lines, reqno-leqno puts eq numbers on right/left
%\documentclass[11pt,draft, reqno,a4paper,psamsfonts]{amsart}
\usepackage{amsmath,amssymb}
\usepackage{graphicx} % replaces epsfig
\newcommand{\comments}[1]{} % to insert a comment
\usepackage{hyperref} % for a URL
% *** CHANGE DIMENSIONS ***
\voffset=-0.3truein % LaTeX has too much space at page top
\addtolength{\textheight}{0.3truein}
\addtolength{\textheight}{\topmargin}
\addtolength{\topmargin}{-\topmargin}
\textwidth 6.0in % LaTeX article default 360pt=4.98''
\oddsidemargin 0pt % \oddsidemargin .35in % default is 21.0 pt
\evensidemargin 0pt % \evensidemargin .35in % default is 59.0 pt
%\parindent=20pt
% *** MACROS ***
\newcommand{\Cal}{\mathcal} % Calligraphic - caps only
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\abs}[1]{\lvert #1 \rvert} % absolute value
\newcommand{\norm}[1]{\lVert #1 \rVert} % norm
\newcommand{\ip}[2]{\langle #1,\, #2\rangle} % ip = inner product
% ======================================
% The remainder of this preface is for sets of problems and their parts.
\newcounter{numb}
\newcounter{bean}
\newenvironment{problems}{\begin{list}{\arabic{numb}.}{\usecounter{numb}
\setlength{\leftmargin}{20pt}
\setlength{\labelwidth}{15pt}
\setlength{\labelsep}{5pt}
\setlength{\itemsep}{ 15.0pt plus 2.5pt minus 1.0pt}
}}{\end{list}}
%The next environment is for the parts a), b), ... of a problem.
\newenvironment{parts}{\begin{list}{\alph{bean}) }{\usecounter{bean}
\setlength{\leftmargin}{20pt}
\setlength{\labelwidth}{15pt}
\setlength{\labelsep}{5pt}
\setlength{\topsep}{0pt}
\setlength{\partopsep}{0pt}
}}{\end{list}}
% ======================================
%==================== END PREAMBLE ==============
\begin{document}
%\pagestyle{empty}
\parindent=0pt
\vspace*{-20pt}
{\bf Math 508, Fall 2014 \hfill Jerry L. Kazdan}
\medskip
\begin{center}
{\large\bf Problem Set 6}
{\sc Due:} Thurs. Oct. 23, 2014. \emph{Late papers will be accepted until
1:00 PM Friday}.
\end{center}
\medskip
{\bf This week}. Please re-read all of Chapter 4 and the first part of
Chapter 5 (through page 108) of the Rudin text.
\smallskip
\comments{
\makeboxbox{\parbox{5.0in}
{\sc Revision Note: } This revised version of Homework Set 3 has
clarifications and hints for problems \# 6, 10, and 11. Problem 7 is
solved in the Class Notes on Compactness and Problem 12 is now a Bonus
Problem.}
}
\vskip15pt
\hrule
\vskip 3pt
\hrule
\medskip
The following short True-False [T/F] questions are exercises that are {\it
not} to be handed-in -- but you should know how to solve them. For
each, either provide a proof or give a counterexample.
\bigskip
\begin{problems}
\item [T/F-1] There is a continuous $f:\R\to \R$ such
that $f(x)=0$ if and only if $x$ is an integer.
\item [T/F-2] If $f:\R\to\R$ is continuous everywhere
and $f(x)=0$ for all rational numbers $x$, then $f(x)=0$ for all real
$x$.
\item [T/F-3] There exists some $x>1$ such that $\frac{x^2+5}{3+x^7}=1$.
\item [T/F-4] The function $f(x):= \abs{x}^3$ is continuous for all
$x\in\R$.
\item [T/F-5] Let $f$, $g$, and $h$ be continuous on
the interval $[0,\,2]$. If $f(0) < g(0) < h(0)$ and $f(2) > g(2) >
h(2)$, then there exists some $c\in [0,\,2]$ such that
$f(c)=g(c)=h(c)$.
\item[T/F-6]
\begin{parts}
\item If $f$ is continuous on $\R$, then $f$ is bounded.
\item If $f$ is continuous on $[0,1]$, then $f$ is bounded.
\item If $f$ is continuous on $\R$ and is bounded, then $f$ attains its
supremum.
\end{parts}
\end{problems}
\hrule
\vskip 3pt
\hrule
\bigskip
{\sc The following problems should be handed-in.}
\begin{problems}
\item Prove that $\cos x$ and $\sin x$ are continuous for all $x\in
\R$. [You may use the usual formulas for $\cos(x+y)$ and $\sin(x+y)$.]
\item Let $f(x):= x^2+4x$. Clearly $\lim_{x\to 0}f(x)=0$. Assuming
that $0<\epsilon<4$, find $\delta > 0$ so that $\abs{x}<\delta$ implies
that $\abs{f(x)}<\epsilon$. Express $\delta$ as a function of
$\epsilon$. [You are not asked to find the {\it best} $\delta$.]
\item Prove that there exists some $x\in[1,2]$ such that
$x^5 + 2x+5 =x^4+10$.
\item Show that at any time there are at least two diametrically
opposite points on the equator of the earth with the same temperature.
Generalize.
\item Construct a function $f$ with the property that there
are sequences $a_n$ and $b_n$ converging to zero such that $f(a_n)$
converges to zero but $f(b_n)$ is unbounded.
\smallskip
Does there exist such a function $f$ that is continuous at $x=0$?
\item Let $f(a,n):= (1+a)^n$, where $a$ and $n$ are positive.
\begin{parts}
\item For constant $a$, how does $f(a,n)$ behave as $n\to\infty$?
For constant $n$, how does $f(a,n)$ behave as $a\to 0$?
\item Let $L\ge 1$ be a given real number. Prove that there exists a
sequence $a_n\to 0$ and $f(a_n,n)\to L$ as $n\to\infty$. In other
words, depending on the choice of $a_n$, the function $f$ may approach
any value.
\end{parts}
\item Which of the following functions are uniformly continuous on
$[0,\,\infty)$ -- and why (or why not)?
\smallskip
a). $f(x)= x\sin x$, \qquad b). $g(x)=e^x$, \qquad
c). $h(x)=\frac{1}{1+x}$
\item Show that $f(x):=\sqrt{x}$ is continuous for all $x \ge 0$. Is
it uniformly continuous there?
\item If $(X,\,d_1)$ any $(Y,\,d_2)$ are two metric spaces (the metrics
are $d_1$ and $d_2$), these metric spaces are called {\it homeomorphic} if
there is a continuous bijection $f:X \to Y$.
\begin{parts}
\item Prove that $[0,1]$ and $\R$ are {\it not} homeomorphic.
\item Prove that $\R$ and $00\}$ are homeomorphic.
\item Prove that $(-1,\,1)$ and $\R$ are homeomorphic.
\end{parts}
\item Let $f(x):=x\sin (1/x)$ for $x\ne0$ while $f(0):=0$.
\begin{parts}
\item Prove that $f$ is continuous for all real $x$.
\item Is $f$ uniformly continuous for $x\in [0,\, 2/\pi]$? Why?
\item Is $f$ uniformly continuous for all real $x$? Why?
\end{parts}
\item Consider $\R^n$ with the Euclidean norm $\abs{x}_2$ and let
$\norm{x}$ be any norm on $\R^n$.
\begin{parts}
\item Let $f(x):\R^n \to \R$ be the function $f(x):=\norm{x}$. Show
that $f$ is continuous at every point of $\R^n$.
\item Show these norms are {\it equivalent} in the sense
that there are constants $c_1>0$, $c_2>0$ such that for any $x\in \R^n$
$$
c_1\abs{x}_2 \le \norm{x} \le c_2\abs{x}_2.
$$
[{\sc Suggestion: } Look at the function
$f(x):=\norm{x}/\abs{x}_2$ on the unit sphere $\abs{x}_2=1$].
\end{parts}
\item Let $f(x)$ be a continuous real-valued function with the property
$$
f(x+y) = f(x)+ f(y)
$$
for all real $x$, $y$. Show that $f(x) = cx$ for some constant $c$.
\item {} [Partly from Rudin, p. 99 \# 8]. Let $E\subset\R$ be a set and
$f:E\to\R$ be uniformly continuous.
\begin{parts}
\item If $E$ is a bounded set, show that $f(E)$ is a bounded set.
\item If $E$ is not bounded, give an example showing that $f(E)$ might
not be bounded.
\end{parts}
\item If $f:\R \to \R$ is uniformly continuous on \emph{all} of $\R$,
show there are constants $a$ and $b$ so that
$$
\abs{f(x)} \le a + b\abs{x}.
$$
\end{problems}
%\comments{
\vskip 20pt
\begin{center}{\large \bf Bonus Problem}
[Please give this directly to Professor Kazdan]
\end{center}
\begin{problems}
\item [B-1] [Rudin, p. 98 \# 3]. Let $\cal M$ be a metric space and
$f:\cal{M}\to \R$ a continuous function. Denote by $Z(f)$ the {\it
zero set of $f$}. These are the points $p\in\Cal M$ where $f$ is
zero, $f(p)=0$.
\begin{parts}
\item Show that $Z(f)$ is a closed set.
\item {} [See also Rudin, p. 101 \#20] Given {\it any} set $E\in\Cal M$,
the distance of a point $p$ to $E$ is defined by
$$
h(p):= \inf_{z\in E}d(p,z).
$$
Show that $h$ is a uniformly continuous function.
\item Use the previous part to show that given any \emph{closed} set
$E\in\Cal M$, there is a continuous function that is zero on $E$ and
positive elsewhere.
\end{parts}
\item [B-2] [Rudin, p. 99 \# 13 or \#11, see also p. 98 \#4]
\emph{extension by continuity} Let $X$ be a metric space, $E\subset X$
a dense subset, and $f:E\to\R$ a uniformly continuous function. Show
that $f$ has a unique continuous extension to all of $X$. That is,
there is a unique continuous function $g:X\to\R$ with the property
that $g(p)=f(p)$ for all $p\in E$.
In your proof, show where it fails if you tried to apply your
procedure to extend the function $f(x):= \sin (1/x)$ from $E:=\{0