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{\bf Math 165: Honors Calculus I}
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{\bf Quiz 9}\quad {\sl Nov.~12, 1998}

\begin{enumerate}
\problemspace
\item
  \begin{enumerate}[a)]
    \item State the \textsc{Boundedness Theorem for Continuous Functions}.
    \answerspace

    \item Use a) to prove that if $f$ is continuous on $[a,b]$ then $f$
    has a maximum on $[a,b]$. (Hint: Let $M = \sup_{[a,b]} f$, $g(x) =
    M - f(x)$, and consider $1/g(x)$.)
    \answerspace

  \end{enumerate}

\newpage
\problemspace
\item Let $f(x)$ be continuous on $[a,b]$ and let $p \in (a,b)$.  Prove
that for any $\varepsilon>0$ there exists a neighborhood of $p$,
$N(p)\subset [a,b]$, such that the span of $f$ on $N(p)$ is
$<\varepsilon$.
\answerspace
\answerspace

\item Suppose $f(x)$ and $g(x)$ are continuous functions on $[a,b]$
satisfying $f(x_1) < g(x_2)$ for any $x_1, x_2 \in [a,b]$. Prove that
there is a \emph{constant} $c$ such that $f(x) < c < g(x)$ for all $x \in
[a,b]$.
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