% LaTeX file -- AM/PM 331, Fall 2011, Assignment 1

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\begin{center}
\textbf{\large AMath/PMath 331 \\ Assignment 3\\
Due Friday October 14}
\end{center}
\vskip 2em 

\begin{Ex}

\item Consider a function defined on $\bR^2$ by
\[
 f(x,y) = \begin{cases}
            0 &\qif y\le0\\
            0 &\qif y\ge x^2\\
            \sin \big( \frac{\pi y}{x^2} \big) &\qif 0 < y < x^2
            \end{cases} .
\]
\begin{parts}
\item Show that $f$ is continuous on $\bR^2\setminus\{(0,0)\}$.

\item Show that $f$ is not continuous at the origin.

\item
Show that the restriction of $f$ to any straight line through
the origin is continuous.
\end{parts}

\item Suppose $f: \bR^n \to \bR^+$ is a positive continuous function 
such that $\dlim_{\|\Bx\|\to\infty} f(\Bx) = 0$.
Prove that $f$ attains its maximum value. \\
\hint there is a large $R$ so that $f(\Bx) < f(\boldsymbol{0})/2$
when $\|\Bx\|>R$.

\item Prove that $4\sin x + 3 \cos x = x$ has at least three real solutions.

\item Let $f:\bR^2 \to \bR$ be a continuous function  such that
\[ f(x,y) = f(x+2,y) = f(x,y+5) \qforal (x,y) \in \bR^2 .\]

\begin{parts}
\item Prove that $f$ attains its maximum and minimum values.\\
\hint why is it enough to consider $(x,y) \in [0,2]\times [0,5]$?

\item Prove that $f$ is uniformly continuous.\\
\hint why is it enough to consider $(x,y) \in [-2,2]\times [-5,5]$?
\end{parts}

\item
\begin{parts}
\item Show that $f(x) = \sqrt x$ is uniformly continuous on $[1,\infty)$.
\item Show that $f(x) = \sqrt x$ is uniformly continuous on $[0,2]$.
\item Show that if $0\le x < y \le 4^{-n-1}$, then
$f(y) - f(x) > 2^n(y-x)$. \\ \hint Mean Value Theorem.
\item \textbf{Don't hand in---just think about it.} Why doesn't (c) contradict (b)?
\end{parts}



\item \textbf{Bonus.} 
 Let $f$ be a continuous function from the closed ball in
$\bR^2$, namely $\ol{B}=\{ x \in \bR^2 : \|x\| \le 1 \}$, into
$\bR$.  Show that $f$ cannot be not one to one.


\end{Ex}
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