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{\bf Math 165: Honors Calculus I}
\hfill\makebox[2.5in][r]{Name:\hrulefill}

{\bf Quiz 9}\quad {\sl Nov.~9, 1995}

\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\answerspace

  \end{enumerate}


\newpage
\item Let $f(x)$ be continuous on $[a,b]$ and suppose $0 < f(x) < 1$ for all 
$x \in [a,b]$. Prove that there is a positive integer $n$ such that
\[
  \frac{1}{n} \le f(x) \le 1 - \frac{1}{n}
\]
for all $x \in [a,b]$
\answerspace\answerspace


\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 neighbohood of $p$,
$N(p)\subset [a,b]$, such that the span of $f$ on $N(p)$ is
$<\varepsilon$.
\answerspace\answerspace


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