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\begin{document}






\topmatter
\title Math 335, Test 3, Fall (1993) \endtitle
\endtopmatter

The test will be 50 minutes in length.  Please write your name on the
cover of your blue book and write the solutions inside; start the
solution to each problem on a new page.  The problems are worth
20~points apiece; in the case of a problem with multiple sections, the
credit will be divided equally between the parts unless otherwise
indicated.  This test is being administered under the provisions of
the Honor Code. Your work should be your own, and you should not make
use of any outside material (texbooks, notes) during the test.  What
you write should be neat, grammatical, clear and concise. Good luck.


%\define\prob#1{\par\medskip\noindent{\bf#1.}\ }
\define\prob#1{\par\medskip\noindent\hbox to 3ex{\bf#1.\hfil}}
%\define\fakeprob#1{\par\noindent\phantom{{\bf #1.}}\ }
\define\fakeprob#1{\par\noindent\hbox to 3ex{\hfil}}
\define\ritem#1{\item"(#1)"}
\define\dom{\operatorname{dom}}
\def\ourstrut{\vrule width 0.0pt depth 0.0pt height 15.0pt\relax}

\prob1
Let $f$ be a continuous real-valued function on the closed interval
$[a,b]$.  Prove that $f$ is a bounded function.



\prob2 Determine whether or not the following functions are {\sl
uniformly continuous} on the indicated sets. Justify your answers.
\roster
\ritem a $x\in(-\pi,\pi),\quad f(x)=\cases\cos(1/x)&x\ne0\\
                          0&x=0\endcases$
\ritem b \ourstrut$x \in(1,2),\quad f(x)=(\sqrt x - 1)/(x-1)$
\ritem c $x\in[-1,1],\quad f(x)=\vert x\vert$
\ritem d $x\in(0,1),\quad f(x)=1/x$
\endroster

\prob3 
Prove that if $f$ is continuous at $x_0$ and $g$ is continuous at
$f(x_0)$ then the composite function $g\circ f$ is continuous at
$x_0$.


\prob4 
Give an example of a series $\sum a_n$ such that $\sum a_n$ converges
but $\sum \vert a_n\vert$ does not converge.  Justify your answer.

\prob5 (a) State the Intermediate Value Theorem. (b) Let $f$ be the
function given by the formula $f(x)=x^5+9x^3+6x$.  Show that for any
real number $y$ there is at least one real number $x$ such that
$f(x)=y$. (Hint: Observe that $\lim_{x\to\infty}f(x)=\infty$ and
$\lim_{x\to-\infty}f(x)=-\infty$.)


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