\documentclass[letterpaper,12pt]{exam}
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% Edit This:
\def\classtitle{Real Analysis}
\def\classnumber{Math 70100}
\def\examtitle{Homework 7}
\def\duedate{November 4, 2019}

\def\skippage{{\ifprintanswers \else \newpage \fi}} % new page if not printing answers


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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 {\em (Folland 4.7.68)}
For a space $X$, let $C(X)$ denote the continuous
functions from $X$ to $\R$ equipped with the uniform norm.
Let X and Y be compact Hausdorff spaces. 
Show that the algebra generated by functions of the form 
$f(x, y) = g(x)h(y)$, where $g \in C(X)$ and $h \in C(Y)$
is dense inside of $C(X \times Y)$. 

\begin{sol}
\end{sol}

\question {\em (Folland 4.7.69)}
Let $A$ be a nonempty set, and let $X = [0, 1]^A$. Show that the algebra generated by the
coordinate maps $\pi_a : X \to [0, 1]$ and the constant function 
${\mathbf 1}$ is dense in $C(X)$.

\begin{sol}
\end{sol}

\question {\em (Rephrased Lang III \S 4 \# 19)}
Let $\R_{\geq 0}=\{x \in \R~:~x \geq 0\}$, and let $C_0(\R_{\geq 0})$ denote the
continuous real-valued functions on $\R_{\geq 0}$ which vanish at infinity.
Prove that $C_0(\R_{\geq 0})$ is the uniform closure of the collection of all functions
of the form $e^{-x} p(x)$, where $p$ is a polynomial.
({\em Lang's Hint; note he phrases this question differently:} First show that you can
approximate $e^{-2x}$ by $e^{-x} q(x)$ for some polynomial q(x), by using Taylor's formula with remainder. 
If $p$ is a polynomial, approximate $e^{-nx} p(x)$ by $e^{-x} q(x)$
for some polynomial $q$.)

\begin{sol}
\end{sol}

\question (Rudin \S 9.6) Define
$$f(x,y)=\begin{cases}
0 & \text{if $(x,y)=(0,0)$,}\\
\frac{xy}{x^2+y^2} & \text{otherwise.}\\
\end{cases}$$
Prove that the partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ exist everywhere, although $f$ is not continuous at $(0,0)$.

\begin{sol}
\end{sol}

\question (Rudin \S 9.7) Suppose that $f$ is a real-valued function defined in an open set $E \subset \R^n$, and that
the partial derivatives $D_1 f, \ldots, D_n f$ are bounded in $E$. Prove that $f$ is continuous in $E$.

\begin{sol}
\end{sol}

\question (Pugh \S 5.22) If $Y$ is a metric space and $f : [a, b] \times Y \to \R$ is continuous, show that
$$F(y) = \int_a^b f(x,y)~dx$$
is continuous. 

\begin{sol}
\end{sol}

\end{questions}
\end{document}
