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\topmatter
\title Math 336, Test 1, Spring (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.

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 Prove that {\sl the uniform limit of continuous functions is
continuous}. More precisely, let $(f_n)$ be a sequence of functions on
a set $S\subset\Bbb R$, suppose that $f_n\to f$ uniformly on~$S$, and
suppose that $S=\operatorname{dom}(f)$. Assume that each $f_n$ is continuous at
$x_0$ in~$S$. Prove
 that $f$ is continuous at $x_0$.

\prob2 Show that the series
$$\sum_{n=1}^\infty \frac{\cos nx}{2^n}$$
converges uniformly on $\Bbb R$ to a continuous function.

\prob3 Let $(f_n)$ be the sequence of functions on $\Bbb R$ given by
$f_n(x)=x/n$. 
\roster
\ritem 1 (6 pts) Determine the function $f$ with domain $\Bbb R$ which is the
pointwise limit of the sequence $(f_n)$.
\ritem 2 (8 pts) Show that the sequence $(f_n)$ {\sl does not} converge uniformly to~$f$.
\ritem 3 (6 pts) Let $g_n$ be the restriction of $f_n$ to the interval $[0,1]$
and let $g$ be the restriction of $f$ to $[0,1]$. Show that the
sequence $(g_n)$
converges uniformly to~$g$.
\endroster

\prob4 Let $f$ be the function on $\Bbb R$ given by $f(x)=x+\vert x
\vert$. Show that $f$ is not differentiable at~$x=0$.

\prob5 Let $f$ be a function which is defined and differentiable on
$\Bbb R$. Suppose that $f(0)=1$, and that $f'(x)<2$ for all $x>0$. Use
the Mean Value Theorem to show that $f(x)<2x+1$ for all $x>0$.

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