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\title{Math 112 Problem Set 7}
\author{Due Wednesday, March 28, 2018}

\maketitle

{\bf Core problems:}
\begin{enumerate} 
	\item It will be useful to have some rules of inverse images of functions.  To set notation, if $f:X \to Y$ is a function and $A \subset X$ and $B \subset Y$, then set
	\begin{eqnarray*}
	f^{-1}(B) &=& \{x \in X \mid f(x) \in B\}\\
	f(A) &=& \{f(x) \in Y \mid x \in A\}
	\end{eqnarray*}
	Show that, for any $B, B' \subset Y$,
	$$f^{-1}(B\cap B') = f^{-1}(B)\cap f^{-1}(B')$$
	Does this hold for images instead of inverse images? I. e., given $A, A' \subset X$, do we have
	$$f(A \cap A') = f(A) \cap f(A)$$
	Prove or find a counterexample.

\item Show that, for any $B, B' \subset Y$,
	$$f^{-1}(B\cup B') = f^{-1}(B)\cup f^{-1}(B')$$
	Does this hold for images instead of inverse images? I. e., given $A, A' \subset X$, do we have
	$$f(A \cup A') = f(A) \cup f(A)$$
	Prove or find a counterexample.

	\item Given $f: \R^n \to \R^m$, we can think of $f$ as a vector of $m$ functions from $\R^n$ to $\R$, that is, write $f = (f_1, \dots, f_m)$ where each $f_k :\R^n \to \R$ and, for every $\vec{x} \in \R^n$, 
	$$f(\vec{x}) = \left(f_1(\vec{x}), \dots, f_m(\vec{x})\right)$$
	Given $\vec{a} \in \R^n$, show that $f$ is continuous at $\vec{a}$ iff each $f_k$ is continuous at $\vec{a}$.

	\item Show that $f(x) = \sqrt{x}$ is continuous at any $a > 0$.

	\item Let $f:\R^2 \to \R$ be continuous.  Show that
	$$\{(x,y) \in \R^2 \mid 0 \leq f(x, y) \leq 1\}$$
	is closed. (4.1, \#3)

	\item The following function is sometimes called Dirichlet's function:
	$$f(x) = \begin{cases} 1 & x  \text{ is rational}\\
	0 & \text{otherwise}\end{cases}$$
	Show that $f$ is discontinuous everywhere.
	\item Define maps $s:\R^n \times \R^n \to \R^n$ and $m:\R\times \R^n \to \R^n$ as $s(\vec{x}, \vec{y}) = \vec{x}+\vec{y}$ and $m(r, \vec{x}) = r\vec{x}$.  Show that these mappings are continuous.(4, \#8)
	\item Let $f:\R\to \R$ be increasing and bounded above.  Show that 
	$$\lim_{x \to 0^+}f(x)$$
	exists. (4, \#35)
	

\end{enumerate}



{\bf Niche problems:}
\begin{enumerate}
	
	\item In proving sequences converge, we often let an $\epsilon>0$, do some stuff, and end up showing the desired absolute value is less than some nice function of $\epsilon$, like $\frac{2\epsilon}{3}$.  We're basically done, but formally we have to go back and pick another epsilon.  The following problem tells us we don't need to worry about this any more!
	
	Let $\{x_n\}\subset \R$ be a sequence and $x \in \R$.  Then $x_n \to x$ iff there is a function $f:(0, \infty) \to (0, \infty)$ such that 
	\begin{itemize}
		\item $\lim_{x \to 0^+} f(x) = 0$; and
		\item for every $\epsilon>0$, there is $N$ such that $n>N$ implies $|x-x_n| < f(\epsilon)$.
	\end{itemize}
	
	\item The following function is sometimes called the raindrop function:
	$$f(x) = \begin{cases} \frac{1}{q} & x \neq 0 \text{ is rational and written in lowest terms as }\frac{p}{q}\\
	0 & \text{otherwise}\end{cases}$$
	Show that $f$ is discontinuous at each nonzero rational and continuous at each irrational and 0.

	\item Given $f:\R \to \R$, set $\Gamma \subset \R^2$ to be the graph of the function, that is,
	$$\Gamma = \{(x, f(x))\mid x \in \R\}$$
	Show that $f$ is continuous iff $\Gamma$ is connected.

	\item  Recall that last problem set, we defined an equivalence relation $\sim$ on the set $\cC$ of rational Cauchy sequences, and then defined addition, multiplication, and order on $\cC/\sim$, the set of equivalence classes.  Show that the turns $\cC/\sim$ into a complete ordered field.
	(As before, this is worth 20 points, replaces 2 Core problems, and counts as 2 Niche problems)
\end{enumerate}

{\bf Doc Brown problems:}
\begin{enumerate}
	\item Set $A = \{x \in \R \mid \sin x = 0.56\}$.  Show that $A$ is a closed set.  Is it compact? (You can use facts about $\sin$ like that it is continuous., 4.3, \#3)
	\item Give an example of a continuous and bounded function on all of $\R$ that does not attain its max or min. (4.4, \#1)
	\item Prove that there is no continuous map from $[0,1]$ onto $(0,1)$. (4.5, \#5)
\end{enumerate}

For problems from the book, something like 1, \#8 refers to \#8 from the exercises at the end of Chapter 1, while something like 1.3, \#1 refers to \#1 from the exercises at the end of section 1.3.
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