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\title{Math 335, Test 2, Fall  1995}

\begin{document}






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.  Each problem has the point
value 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 (textbooks, 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}
\define\R{\Bbb R}
\define\Q{\Bbb Q}
\define\Limit{\Cal L}

\prob1 (20 pts)
Suppose that $f:S\to \R$ where $S\subset\R$, and suppose that
$a\in\Limit(S)$. Prove that the following two conditions are
equivalent.
\roster
\ritem a $f(x)\to \alpha$ as $x\to a$.
\ritem b For every sequence $(x_n)$ in the set $S\setminus\{a\}$
satisfying $x_n\to a$, we have $f(x_n)\to\alpha$.
\endroster 

\prob2 (20 pts) 
Suppose that $S$ and $T$ are sets of real numbers, that $f:S\to T$ and
that $g:T\to\R$. Suppose that $f$ is continuous at a point $x\in S$
and that $g$ is continuous at the point $f(x)$. Prove that the
composition $g\circ f$ is continuous at the point~$x$.



\prob3 (a) (10 pts) Suppose that~$a$ is a real number. Define what it means for $\lim_{x\to a}f(x)$ to
be~$-\infty$. (b) (10~pts) Suppose that $\alpha$ and~$a$ are real
numbers. Define what it means for $\lim_{x\to a^{+}}f(x)$ to be equal
to~$\alpha$.

\prob4 (20 pts)
Starting with your favorite precise formulation of the notion of ``limit'' (for
instance, the $\epsilon$-$\delta$ formulation), give a careful
argument to show that 
$$\lim_{x\to 1} (x^2+1) = 2\,\,\,.$$


\prob5 
(a) (10 pts) Suppose that $a$ is a point in the domain of a
function~$f$. Give the $\epsilon$-$\delta$ formulation of what it
means for~$f$ to be continuous at~$a$.
(b) (10~pts) Suppose that $f$ is a function whose domain is the set of
all real numbers. Assume that for any two real numbers $x$ and $y$,
the following inequality is true:
$$\vert f(x)-f(y)\vert \le \vert x-y\vert\,\,.$$
Using~(a), show that $f$ is continuous at every point of its domain.


\end{document}



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