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\title{Math 112 Problem Set 8}
\author{Due Wednesday, April 4, 2018}

\maketitle

{\bf Core problems:}
\begin{enumerate} 
	\item The following is a further exploration of the properties of images and inverse images.  Let $f:X \to Y$ and show the following: (this is worth 2 problems)
	\begin{enumerate}
		\item Given $A, A' \subset X$, if $A \subset A'$, then $f(A) \subset f(A')$.  Find a counter-example showing the converse doesn't hold.
		\item Given $A \subset X$, $A \subset f^{-1}(f(A))$.  Find an example showing equality doesn't hold.
		\item Given $B \subset Y$, $f(f^{-1}(B)) = B \cap f(X)$.
	\end{enumerate}
	
	\item Find a continuous function $f$ and an open subset $A$ of its domain such that $f(A)$ is not open. (You can use the functions from calculus if you want, although this is not necessary.)
	
	\item Let $f:[0,1] \to [0,1]$ be continuous.  Show that $f$ has a fixed point; that is, there is $x \in [0,1]$ such that $f(x) = x$. (4.5, \#3)
		
	\item \begin{enumerate}
		\item Find $f:\R^2 \to \R$ such that
		$$\lim_{x\to 0}\lim_{y\to 0} f(x, y) \neq \lim_{y \to 0} \lim_{x\to 0} f(x,y)$$
		although both exist.	
		\item Find $f:\R^2 \to \R$ such that
		$$\lim_{x\to 0}\lim_{y\to 0} f(x, y) = \lim_{y \to 0} \lim_{x\to 0} f(x,y)$$
		but $f$ is not continuous at $0$.
		\item Find $f:\R^2 \to \R$ that is continuous on every line through $(0,0)$ but not continuous at $(0,0)$.  Being continuous on every line means that, for every $m \in \R$, $\lim_{t \to 0}f(t, mt) = f(0,0)$ and $\lim_{t\to 0}f(0, t) = f(0,0)$.
		\end{enumerate}(You may use the various functions from calculus for this exercise; 4, \#14)

	\item Let $K \subset \R^n$ be compact and let $f: K \to \R^m$ be continuous and one-to-one.  Show that $f^{-1}:f(K) \to K$ is continuous.  Find an example that this can fail if $K$ is connected but not compact. (4, \#7)
	
	\item \begin{enumerate}
		\item If $f: \R \to \R$ is continuous and $A \subset \R$ is connected, is $f^{-1}(K)$ connected?
		\item Show that if $f:\R^n \to \R^m$ is continuous on $\R^n$ and $B \subset \R^n$ is bounded, then $f(B)$ is bounded.
	\end{enumerate}(4, \#3)
	
	\item If $f:\R \to \R$ is continuous and $K \subset \R$ is compact, is $f^{-1}(K)$ compact? Prove or find a counter example.

\end{enumerate}



{\bf Niche problems:}
\begin{enumerate}
	
		\item Let $A \subset \R^n$ and $f:A \to \R^m$.  Show that $f$ is continuous iff for every $B \subset A$, 
	$$f\left(cl(B) \cap A\right) \subset cl\left(f(B)\right)$$(4, \#10)
	
	

	
	\item Show that closed intervals $[a, b]$ are the only subsets of $\R$ that are both compact and connected.
	\item We proved in class that continuous functions satisfy the intermediate value theorem.  This exercise shows the converse fails.
	
	Define $f:\R \to \R$ by
	$$f(x) = \begin{cases} 0 & x = 0\\
	\sin \frac{1}{x} & x \neq 0\end{cases}$$
	
	Show that $f$ satisfies the Intermediate Value Property: for any $a < c \in \R$ and $z \in \R$ such that $z$ is between $f(a)$ and $f(c)$, there is a $b \in [a, c]$ such that $f(b) = z$.
	
	Find a value $x_0 \in \R$ such that $f$ is not continuous at $x_0$ (and show this).\\
	
	You can use facts about $\sin$ for this exercise. 
	
	\item We've built complete, ordered fields from Dedekind cuts and from equivalence classes of Cauchy sequences of rationals.  But maybe this seems strange since we like to think of real numbers as, well, numbers.  Specifically, we like to write down numbers as decimals in base 10.  This exercise will attempt to make the point that this is a hard thing to make precise.
	
	A base 10 representation should consist of $(k; a_k, \dots, a_1; \{b_n\})$ where
	\begin{itemize}
		\item $k$ is a nonnegative integer;
		\item each $a_i$ is an integer between $0$ and $9$; and
		\item $\{b_n\}_{n=1}$ is a (infinite) sequence of numbers between $0$ and $9$.
	\end{itemize}
	This is a weird way of writing it, but we imagine that $(k; a_k, \dots, a_0; \{b_n\})$ represents the real number
	$$\sum_{i=0}^k a_i 10^i + \sum_{j=0}^\infty b_i 10^{-i}$$
	\begin{enumerate}
		\item Show that this sequence converges for any choice of a base 10 representation.
		\item Define an ordering $(k; a_k, \dots, a_1; \{b_n\}) \prec (k'; a'_{k'}, \dots, a'_1; \{b'_n\})$ that agrees with this ordering of real numbers.
		\item Can you define addition for these representations?  What about multiplication?  You don't have to succeed here, but at least talk about why this is a very hard task.
	\end{enumerate}
\end{enumerate}

{\bf Doc Brown problems:}
\begin{enumerate}
	\item Give an example of a function defined on $\R$ which is continuous everywhere but fails to be differentiable exactly at $x_1, \dots, x_n$. (4.7, \#1)
	\item Does the mean value theorem apply to $f(x) = \sqrt{x}$ on $[0,1]$? Does it apply to $g(x) = \sqrt{|x|}$ on $[-1, 1]$? (4.7, \#2)
\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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