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\author{Math 25}

\title{Midterm}

\begin{document}

\maketitle
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\section*{Instructions: read carefully}

This midterm is due \textsc{Monday November 3rd at noon}. You should return it
in class Monday, the noon deadline being there to avoid catastrophes
(alarm clock not working, too much candy after Halloween, etc). If you
return the midterm after class, put it in my mailbox. 

\begin{itemize}
\item You \textbf{are} allowed to use: your notes from class, 
anything that's on the course's website or on the CA's website.
\item
You \textbf{are not} allowed to use: (text)books, photocopies,
anything that's on the
web, a calculator or a computer. You are not allowed to discuss 
the midterm with your classmates or anybody else (except your CA's and me). 
\end{itemize}

I have read the above and agree to it:


\vspace{50pt}

Name:\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_
Signature:\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_

Attach this to your solutions. I will only
accept paper (not emailed) solutions. 

\vspace{20pt}
\begin{tabular}{||c||c|c|c|c||c|c|c|c||c|c|c|c||c||}
\hline
{\S} & 1.1 & 1.2 & 1.3 & 1.4 & 2.1 & 2.2 & 2.3 & 2.4 & 3.1 & 3.2 & 3.3 & 3.4
  & Total \\
\hline 
Points & & & & & & & & & & & & & \\
\hline 
Out of & 8 & 8 & 12 & 12 & 8 & 8 & 12 & 12 & 12 & 12 & 12 & 4 & 120 \\
\hline
\end{tabular}
\vspace{20pt}

\section{An application of Taylor's expansion}\label{divc}

In this problem, $a \in \RR$ and $f:\RR\ra \RR$ is a
$C^\infty$ function. The goal of the problem is to prove that $x
\mapsto (f(x)-f(a))/(x-a)$, which is defined on $\RR \setminus \{a\}$,
extends to a $C^\infty$ function on $\RR$. 

\subsection{8pts} Prove that there exists a continuous map $g:\RR\ra\RR$ such that
  for every $x \neq a$, we have \[ g(x) = \frac{f(x)-f(a)}{x-a}. \] 
What is $g(a)$ equal to?
\subsection{8pts} Prove that $g:\RR\ra\RR$ is $C^1$ and find $g'(a)$. 
\subsection{12pts} Assume for this question that $f(a)=f'(a)=\cdots=f^{(n+1)}(a)
  = 0$, where $n \in \ZZ_{\geq 0}$. Prove that  $g:\RR\ra\RR$ is
  $C^n$ and that $g(a)=g'(a)=\cdots=g^{(n)}(a) = 0$.
\subsection{12pts} Return now to the case of a general $C^\infty$ function 
$f:\RR\ra\RR$. Prove that $g:\RR\ra\RR$ is $C^\infty$ 
and compute $g^{(n)}(a)$ for $n \geq 0$.

\section{Derivations of $C^\infty(\RR,\RR)$}

In this  problem, we will use the results from problem \ref{divc}. 
Let $C^\infty(\RR,\RR)$ denote the set of all maps $f:\RR\ra\RR$ which
are $C^\infty$. The goal of this problem
is to find all maps $T: C^\infty(\RR,\RR) \ra
C^\infty(\RR,\RR)$ satisfying the 
following three properties:
\begin{enumerate}
\item $T(f+g)=T(f)+T(g)$;
\item $T(\lambda f)=\lambda T(f)$ if $\lambda \in \RR$;
\item $T(fg)=fT(g)+gT(f)$. 
\end{enumerate}
Observe that $T(\phi)=\phi'$ is one such map.

\subsection{8pts} Let $1$ denote the constant function $x \mapsto 1$. What is $T(1)$? 
\subsection{8pts} For $a\in\RR$, let $\theta_a : \RR \ra \RR$ be the map
$x \mapsto (x-a)^2$ and let $\varepsilon_a=T(\theta_a)$. What is
$\varepsilon_a(a)$? 
\subsection{12pts} Let $\phi \in C^\infty(\RR,\RR)$ be a function such that
$\phi'(a)=0$. Prove that $T(\phi)(a)=0$.
\subsection{12pts} Prove that there exists $f \in C^\infty(\RR,\RR)$ such
that for all $\phi \in C^\infty(\RR,\RR)$, we have $T(\phi)=f \phi'$.

\section{A characterization of ruled functions}
Let $I=[a,b]$ be a closed interval, and let $f:I\ra\RR$ be a
function. We say that $f$ has a left limit at $x \in I$ if there
exists $y^- \in \RR$ such that for every sequence $\{x_n\}_{n \geq 1}$
satisfying $x_n <x$ and $x_n \ra x$, we have $f(x_n) \ra y^-$. The
definition of right limits $y^+$ is analogous.

The goal of this problem is to prove that a function $f:I\ra\RR$ is
ruled if and only if at every $x \in I$, $f$ has a left limit and a
right limit (at $x=a$ we only ask for a right limit and at $x=b$
we only ask for a left limit).

Let $L(I)$ be the set of functions $f:I\ra\RR$ such that 
at every $x \in I$, $f$ has a left limit and a
right limit (at $x=a$ we only ask for a right limit and at $x=b$
we only ask for a left limit).
Let $S(I)$ be the set of step functions $f:I\ra\RR$. If $f:I\ra\RR$ is
a bounded function, let $|f|_I$ denote $\sup_{x \in I} |f(x)|$.

\subsection{12pts} Prove that if $f \in L(I)$, then $f$ is bounded.

\subsection{12pts} Prove that if one defines $d(f,g)=|f-g|_I$ for $f$, $g
\in L(I)$, then $(L(I),d)$ is a complete metric space.

\subsection{12pts} Explain why $S(I) \subset L(I)$. Prove that $(S(I),d)$
is dense in $(L(I),d)$. Hint: prove that if $f \in L(I)$ and
$\varepsilon > 0$, then the set of $x \in I$ such that there exists
$\phi \in S(I)$ satisfying $\sup_{y \in [a,x]} |f(y)-\phi(y)| \leq
\varepsilon$ is both open and closed.
 
\subsection{4pts} Conclude: prove that $L(I)$ is equal to the set of
ruled functions on $I$.
\end{document}