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%==================== END PREAMBLE ==============
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\vspace*{-50pt}
{\bf  Math 361, Fall 2015\hfill Jerry L. Kazdan}
\medskip

\begin{center}
{\large\bf Problem Set 2}
\smallskip

%\fbox{{\sc \large Draft}}
\end{center}

%\comments{
{\sc Due:} Thurs. Sept.~10 in class. [Late papers will be accepted (without penalty) until 1:00 PM Friday.]
%}
\medskip

 Please carefully read Sections 5.1--5.5 and 5.7 in the Marsden-Hoffman text. 

\bigskip

\begin{problems}

\item Prove the {\it Integral Intermediate Value Theorem}: If $f$ is
real and continuous on $[a,b]$, then there exists $c \in (a,b)$ such that
\[ 
\frac{1}{b-a} \int_a^b f(x) \, dx = f(c). 
\]
Also, give an example showing that such a $c$ may not exist if $f$ is
not continuous.

\item {} [Like Marsden-Hoffman p.~316 \#2] Determine which of the following function sequences of functions
converge pointwise or uniformly:
\begin{parts}
\item $f_n(x)=\frac {\sin\,x}n,\,x\in \R$
\item $f_n(x)=\frac 1{1+nx},\,x\in [0,\,1]$
\item $f_n(x)=\frac x{1+nx^2},\,x\in \R$. 
\end{parts}

\item {}[Marsden-Hoffman p.~245 Ex.~4]  Discuss the uniform convergence of
$\sum_{n=1}^\infty 1/(x^2+n^2)$.

\item A continuous function is called {\it piecewise linear} if it
  consists only of straight line segments (see 
\url{https://en.wikipedia.org/wiki/Piecewise_linear_function})

Let $f:[a,b]\to \R$ be a continuous function.  Show that given any
$\epsilon>0$, there is a piecewise linear function $g:[a,b] \to \R$
such that $\abs{f(x)-g(x)} < \epsilon$ for all $x \in [a,b]$.  In
other words, any continuous function on $[a,b]$ can be approximated
``uniformly'' by a piecewise linear function.



\item Give a proof or a counterexample for the following questsions.
\begin{parts}
\item If $f_n\to f$ and $g_n\to g$ pointwise in $C([0,\,1])$. 
is it true that $f_n\cdot g_n\to f\cdot g$ pointwise?   
\item If $f_n\to f$ and $g_n\to g$ uniformly in $C([0,\,1])$,
is it true that $f_n\cdot g_n\to f\cdot g$ uniformly?
\end{parts}


\item Let $f\in C([0,\infty))$ be a continuous function with the
property that $\lim_{x\to \infty}f(x) = c$.  Show that
$$
\lim_{T \to \infty}\frac{1}{T}\int_0^T f(x)\,dx = c.
$$

\item Compute
$\displaystyle \lim_{\lambda \to \infty}\int_0^1 \abs{\sin(\lambda x)}\,dx$.

\vspace*{-1.9cm} 
\item \begin{minipage}[t]{3.9in}
Let $f(x)$ be continuous on the interval $[-1,\,1]$ and
$g_n(x)$ be the sequence of functions in the figure.  Show that 
$$
\lim_{n\to\infty}\int_{-1}^1 f(x)g_n(x)\,dx = f(0).
$$
{\sc Suggestion } First do the case where $f(x)\equiv 1$.
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\end{problems}

%\comments{
\vskip 20pt

\begin{center}{\large \bf Bonus Problem}
			
[Please give this directly to Professor Kazdan]
\end{center}

\begin{problems}

\item [B-1] Let $f(x)$ be a continuous function for $0\le x \le 1$.
Evaluate \ $\ \displaystyle  \lim_{n\to \infty} \int_0^1
nf(x)x^n\,dx.$  (Justify your assertions.)


 \end{problems}
%}


\medskip
[Last revised: \today]
\end{document}

