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{\course}         \hfill {Instructor: Russell Brown}\\ 
{\semester}	  \hfill  {\today }

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Problems 6 to 12 are due  31 January 1996.  Problems 13--16 are due
Wednesday, 7 February.  For class on Monday, 5 February 1996, you
should read section 4 and 5 of Chapter IV on continuous functions on
compact and connected sets.  The following set of problems will keep
us busy for several classes.  Problems 17 to 22 relate to section 4.
The remaining problems relate to section 5.  


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\begin{enumerate}
\setcounter{enumi}{16}

\item Complete the following  without using no, not or never: Let $E$
and $E'$ be metric spaces.
A function $f:E \rightarrow E'$ is not uniformly continuous if $\dots$

\item Chapter 4, problem 13.

\item Chapter 4, problem 16.  Hint:  You might want to show ${\cal C}
f(A) = f ({ \cal C} A)$.


\item Chapter 4, problem 17.

\item Chapter 4, problem 19.

\item 
\begin{enumerate}\item  Let $f_0,\dots , f_N$ be a family of linear
functions\footnote{Assume that each $f_i$ is of the form $a_0+ \sum_{j=1} ^n
a_j x_j $. Strictly speaking these are not linear, but perhaps affine
linear.} from $E^n$ to $\reals$.  Set $ K = \cap _{k=1}^N \{
(x_1,\dots, x_n): f_k(x_1,\dots,x_n) \geq 0\}$.  Show that if $K$ is a
bounded set, then there is a point $(y_1,\dots,y_n)$ which satisfies
$$
f_0(y_1,\dots y_n)  = \sup \{f_0 (x_1,\dots, x_n) : (x_1,\dots,x_n)\in
K\}.$$

\item 
Many of you have seen the result from part a) in another course. How is
this result proven there?
\end{enumerate}

\item
\begin{enumerate}
\item Prove that a polynomial of odd degree on $R$ always has a root.
Hint:   You may assume that the coefficient of highest degree is
1. Then, try to show that for $x$ large and positive, the polynomial
is positive. If $p(x) = x^d+ a_{d-1} x^{d-1}+ \dots + a_1 x + a_0$,
 then  $p(x) $ is positive if $ x > \max (1, |a_0| + |a_1|
+ \dots +|a_{d-1}|)$. Also,  show that for $x$ large and negative, $p(x)
$ is negative. 
\item Chapter 4, problem 27.
\end{enumerate}

\item Chapter 4, problem 25.  

\item Chapter 4, problem 28. Hint: Use a proposition on page 60.

\item Chapter 4, problem 30. (This may be a hard one. I will let you
know when I figure out how to do it.)

\end{enumerate}
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