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\begin{document}
\title{Analysis Seminar: Assignment 1}
\maketitle
Read sections five through seven in Munkres.
\section*{Doing}
I would like you to try to do all the problems in these sections.
This may be an unreasonable expectation, so what I ask that you do is
start with problems 5.2, 5.7, 6.4, 6.7, 6.9, 6.10, 7.3, and the one I
have written below.  After
you have done these, spend the rest of the time you allocate to
problem solving working on the other problems in what ever order makes
sense to you.

\begin{ex}
Suppose that $$A \subset \R^{n}$$ and that $$f : A @>>> \R^{m}$$
is differentiable at every point of $$A$$.  If one thinks of $$Df({\bf a})$$ as an $$m$$ by $$n$$ matrix and one thinks of the
collection of $$m$$ by $$n$$ matrices as $$\R^{nm}$$ then $$Df$$ may be thought of as a function $$A @>>> \R^{nm}$$.  Show that $$f$$ is $$C^{1}$$ if and only if $$Df$$ is continuous.  State and
prove a generalization to $$C^{r}$$ for $$r \geq 0$$.  (I am
asking you to supply a reasonable definition of $$C^{0}$$ here.)
\end{ex}
\section*{Writing}
Pick three or more of the high priority problems to write up
carefully and hand in to me.
\section*{Presentations}
\begin{pres}[Jeremy]
A careful solution to exercise 5.2 with illustrative examples using
Mathematica and Spiderwoman.
\end{pres}
\begin{pres}[Sonya]
Elucidation of Theorem 6.3.
\end{pres}
\begin{pres}[James]
Elucidation of Theorem 7.1.
\end{pres}
\end{document}