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\topmatter
\title Math 335, Test 3, Fall (1992) \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.

Note: all functions below are {\sl real-valued}.

\define\prob#1{\par\medskip\noindent{\bf#1.}\ }
\define\ritem#1{\item"(#1)"}
\define\dom{\operatorname{dom}}
\def\ourstrut{\vrule width 0.0pt depth 0.0pt height 25.0pt\relax}

\prob1 Consider the series
$\displaystyle\sum_{i=1}^\infty\frac{(-1)^n}{n^{1/10}}$.
\roster\ritem a Does this series
converge? \ritem b Does this series  converge absolutely? 
\endroster In each case justify your answer.

\prob2 
(a) [5 pts] Give the definition of what it means for a function~$f$ to
be {\sl continuous}. 
\roster
\ritem b [10 pts]  Let $f$ be a continuous 
function with domain $(a,b)$.  Suppose that
$f(r)=0$ for each rational number $r\in(a,b)$. Show that $f(x)=0$ for
all $x\in(a,b)$.
\ritem c [5 pts] Let $f$ be the function with domain $(a,b)$ defined by
letting $f(x)=0$ for rational numbers~$x$ and $f(x)=1$ for irrational
numbers. Show that $f$~is not continuous.
\endroster

\prob3 Suppose that $f$ is a continuous function on the closed
interval~$[a,b]$.  Prove that $f$~is bounded.

\prob4 (a) State the Intermediate Value Theorem. (b) Suppose that
$f$~is a continuous function on~$\Bbb R$ and that
$f(a)f(b)<0$ for some $a$,~$b\in\Bbb R$. Show that there exists a number
$x$~between $a$~and $b$ such that $f(x)=0$.

\prob5  Determine whether or not the following functions are uniformly
continuous on the specified sets.
\roster
\ritem a $\displaystyle f(x)=\frac{x-1}{x+2}$ on $[0,1]$.
\ritem b \ourstrut$\displaystyle f(x)=\sin(\frac1{x^2+1})$ on $(0,1)$.
\ritem c \ourstrut$\displaystyle f(x)=\max(-1, -1/x)$ on $(0,1)$.
\ritem c \ourstrut$\displaystyle f(x)=\sum_{i=0}^\infty x^i$ on $(-1,0]$. (Note that the
series converges for $x$~in this interval.)
\endroster

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