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\noindent Quiz 2

\noindent Math 336, Winter `00

\vspace*{.2in}
\noindent 1.  Let $\mathcal{M}$ be a set and 
$\rho:\mathcal{M}\times\mathcal{M}\to\mathbf{R}$ a function.  What must 
$\rho$ satisfy in order to be a metric for $\mathcal{M}$?  
(5 points)
\vspace*{2.5in}

\noindent 2.  Following are four sequences of continuous functions on
$[0,1]$.  One sequence converges with respect to the metric 
$\rho(f,g) = ||f-g||_\infty$.  One of the other two
converges in the metric $\rho(f,g) = \int_0^1|f(x)-g(x)|\,dx$.
Identify these sequences and their limits. (5 points)
\bigskip

\begin{enumerate} 
\item $f_n(x) = n(1-x)^n$
\bigskip
\item $f_n(x) = \left\{\begin{array}{l} 1 \text{ if } x \geq 1/n \\
                                        nx \text{ if } x < 1/n
                       \end{array}\right.$
\bigskip 
\item 
$f_n(x) = \sin(2\pi nx)$
\bigskip
\item $f_n(x) = .\left\{\begin{array}{l} x/n \text{ if } x \leq 1/2 \\
                                        (1-x)/n \text{ if } x \geq 1/2
                       \end{array}\right.$
\end{enumerate}


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