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%\input{/home/pat/ownCloud/math/my_papers/symbols.tex}

% Edit This:
\def\classtitle{Real Analysis}
\def\classnumber{Math 70100}
\def\examtitle{Homework 8}
\def\duedate{November 11, 2019}

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\begin{document}
\noindent
{\large \bf \classnumber: \classtitle (Prof. Hooper) \hfill Due \duedate}\\
{\large \bf \examtitle}\\

\ifprintanswers 
\noindent
{\bf Solutions:}
\else
\noindent
%{\bf Instructions:}
%Think about how to write clear and concise answers. Write the solution in your own words.
\fi

\begin{questions}
\question(Pugh \S 5 \#20) Assume that $U$ is a connected open subset of $\R^n$ and $f : U \to \R^m$ is differentiable everywhere on $U$. If $(Df)_p = 0$ for all $p \in U$, show that $f$ is constant.

\begin{sol}
\end{sol}

\question(Pugh \S 5 \#30) Consider the equation
$$x e^y + y e^x = 0.$$
\begin{parts}
\part Observe that there is no way to write down an explicit solution $y = y(x)$
in a neighborhood of the point $(x_0 , y_0) = (0, 0)$.
\part Why, nevertheless, does there exist a $C^\infty$ solution $y = y(x)$ near
(0, 0)?
\part What is its derivative at $x = 0$?
\part What is its second derivative at $x = 0$?
\part What does this tell you about the graph of the solution?
\end{parts}

\begin{sol}
\end{sol}

\question(Pugh \S 5 \#33) Observe that $Y = Inv~X$ solves the implicit function problem
$$F(X, Y) - I = 0,$$
where $F (X, Y ) = X \cdot Y.$ Assume it is known that $Inv$ is smooth and use the
Chain Rule to derive from this equation the formula for the derivative of $Inv$.

\begin{sol}
\end{sol}

\question(Rudin \S 9 \#16) Show that the continuity of $f'$ at the point $a$ is needed in the inverse function
theorem, even in the case $n = 1$: If 
$$f(t)=\begin{cases}
t + 2 t^2 \sin(\frac{1}{t}) & \text{if $t \neq 0$} \\
0 & \text{if $t=0$,}
\end{cases}$$
then $f'(0)=1$, $f'$ is bounded in $(-1,1)$, but $f$ is not one-to-one in any neighborhood of $0$.

\begin{sol}
\end{sol}

\question(Fall 2010 qual) For $t \in \R$, define a sequence $s_1, s_2, \ldots$ by setting $s_1=t$, and for $n \geq 1$, $s_{n+1}=\cos s_n$. Show that $\{s_n\}$ converges to a value independent of $t$. ({\em Hint:} You may find it helpful to examine the relationships between the graphs of $y=\cos x$ and $y=x$.)

\begin{sol}
\end{sol}

\question(Fall 2013 qual) 
Suppose $f: \R \to \R$ is differentiable. If $\int_{-\infty}^\infty |f(x)|~dx < \infty$ and $\int_{-\infty}^\infty |f'(x)|~dx < \infty$, prove that $f(x) \to 0$ as $x \to \infty$. 

\begin{sol}
\end{sol}

\end{questions}
\end{document}
