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\title{Homework 9 for 18.100B/C, Fall 2010\\
Due Thursday 18 November}
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%\author{Richard Melrose}
%\address{Department of Mathematics, Massachusetts Institute of Technology}
%\email{rbm@math.mit.edu}
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As usual, homework is due in 2-108 by 11AM on Thursday 18 November, or by
email before 5PM on the same day.

\begin{enumerate}
\item A function $f:[a,b]\longrightarrow \mathbb{R}$ is said to
  be `Lipschitz continuous' (or just `Lipschitz') if there exists a
  constant $A$ such that
$$
|f(x)-f(y)|\le A|x-y|\ \forall\ x,y\in[a,b].
$$
Show that if $f:[a,b]\longrightarrow \mathbb{R}$ is differentiable and
  $f':[a,b]\longrightarrow \mathbb{R}$ is bounded then $f$ is Lipschitz. 

\item Suppose that $g:[0,1]\longrightarrow\mathbb{R}$ is a
  Lipschitz function and that $f:[0,1]\longrightarrow[0,1]$ is a
  differentiable functions satisfying
$$
f'(x)=g(f(x))\ \forall\ x\in[0,1].
$$
Show that $f':[0,1]\longrightarrow \mathbb{R}$ is Lipschitz.
\item Rudin Chap 7, No 1. Prove that every uniformly convergent sequence of
  bounded functions is uniformly bounded.
\item Rudin Chap 7, No 2. If $\{f_n\}$ and $\{g_n\}$ are sequences of
  functions defined on a set $E$ and each converges uniformly on $E$ prove
  that $\{f_n+g_n\}$ converges uniformly on $E.$ If in addition both
  sequences consist of bounded functions, prove that $f_ng_n$ converges uniformly
  on $E.$
\item Rudin Chap 7, No 6. Prove that the series of functions
$$
\sum\limits_{n=1}^{\infty}(-1)^n\frac{x^2+n}{n^2}
$$
on the real line converges uniformly on any bounded subset of the reals but
does not converge absolutely for any value of $x.$

\end{enumerate}

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