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\begin{document}

\hbox to \hsize{\textbf{Math 331\hfill Homework 6}}
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\centerline{\textit{This homework is due by 11:59 PM on Thursday, October 15.}}

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\def\ds{\displaystyle}


\begin{problem}  % problem 1
Let $\{x_n\}_{n=1}^\infty$ be a convergent sequence in a metric
space.  Show that its limit is unique.  That is, prove the following statement:
if $\{x_n\}_{n=1}^\infty$ converges to $y$ and 
$\{x_n\}_{n=1}^\infty$ converges to $z$, then $y=z.$
[Hint: You can just imitate the proof for a real-valued function of a 
real variable.]
\end{problem}

\begin{answer}
%You can put your answer here for problem 1, 3 points
\end{answer}




\begin{problem}  % problem 2
Let $X$ be any set, and Consider the metric space $(X,\delta)$ where $\delta$ is the
discrete metric, $\delta\colon X\times X\to \R$ by $\delta(a,b) = 
\begin{cases} 0&\text{if }a=b\\ 1&\text{if }a\not=b\end{cases}\;.$
Suppose that $\{x_i\}_{i=1}^\infty$ is a \textbf{convergent} sequence in the
metric space $(X,\delta).$  Show that there is a number $N$ such
that $x_N=x_{N+1}=x_{N+2}=\cdots$.  [Hint: The number is the limit of the sequence.]
(We say that the sequence is
``eventually constant.'')
\end{problem}

\begin{answer}
%You can put your answer here for problem 2, 3 points
\end{answer}




\begin{problem}  % problem 3
Let $(X,\delta)$ be the discrete metric space from the previous exercise,
and let $(M,d)$ be any metric space.  Show that \textbf{any} function
$f\colon X\to M$ is continuous.  (There are at least three
possible proofs: using the definition of continuity, using 
Theorem~3.1, or using Theorem~3.3 from Metric Spaces Section~3.)\end{problem}

\begin{answer}
%You can put your answer here for problem 3,  points
\end{answer}




\begin{problem}  % problem 4
Define $f\colon \R\to\R$ by $f(x)=\begin{cases}x&\text{ if $x \le 1$}\\x+1&\text{ if $x > 1$}\end{cases}.$
We know from calculus that $f$ is not continuous at 1, so it must fail the continuity tests
in Theorems~3.1 and~3.3 from Metric Spaces Section~3.
Find a sequence $\{x_i\}_{i=1}^\infty$ that converges to 1, but  $\{f(x_i)\}_{i=1}^\infty$
does not converge to $f(1).$  And find an open subset $\mathcal O$ of $\R$ such that $f^{-1}(\mathcal O)$ is not open.
(The values of the limits and of $f^{-1}(\mathcal O)$ that you look at should be clear, and you
do not need to prove that the values that you assert are correct.)
\end{problem}

\begin{answer}
%You can put your answer here for problem 4, 3 points
\end{answer}




\begin{problem}  % problem 5
[Textbook problem 3.1.3a]
Even though $|x|$ is not differentiable at 0, show that the function $f(x)=x|x|$ is
differentiable at~0, and find $g'(0)$.
\end{problem}

\begin{answer}
%You can put your answer here for problem 5, 3 points
\end{answer}




\begin{problem}  % problem 6
[Textbook problem 3.2.8]
Use mathematical induction, the product rule, and the fact that $\frac{d}{dx}x=1$ 
to prove that $\frac{d}{dx}x^n = nx^{n-1}$ for all $n\in\N$.
\end{problem}

\begin{answer}
%You can put your answer here for problem 6, 3 points
\end{answer}




\begin{problem} % problem 7
Recall that $f$ satisfies a Lipschitz condition if there is a constant $M$ such that
$|f(b)-f(a)|\le M|b-a|$ for all $a,b$,
and any function that satisfies a Libschitz condition is uniformly continuous.
Let $f$ be a function that is differentiable everywhere, and suppose $f'(x)\le M$
for all $x$, where $M$ is some constant.  Use the Mean Value Theorem to prove that
$|f(b)-f(a)|\le M|a-b|$ for all $a,b$.  Conclude that $f$ is uniformly continuous.
\end{problem}

\begin{answer}
%You can put your answer here for problem 7, 3 points
\end{answer}




\begin{problem}  % problem 8
[Textbook problem 3.3.10]
A \textbf{fixed point} of a function is a point $d$ such that $f(d)=d$.  Suppose that
$f$ is differentiable everywhere and that $f'(x)<1$ for all $x$.  Show that there can
be at most one fixed point for $f$. [Hint:  Suppose that $a$ and $b$ are two fixed points of $f$.
Apply the Mean Value Theorem to obtain a contradiction.]
\end{problem}

\begin{answer}
%You can put your answer here for problem 8, 3 points
\end{answer}





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