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\title{Math 112 Problem Set 11}
\author{Due Wednesday, April 25, 2018}

\maketitle

{\bf Core problems:}
\begin{enumerate}
	\item Prove that a sequence $f_k : A \to \R^m$ converges pointwise iff, for each $x \in A$, $\{f_k(x)\}_k$ is a Cauchy sequence. (5, \#6)
	\item Does pointwise convergence of continuous functions to a continuous limit imply uniform convergence on that set? (5, \#8)
	\item From the sequences in (7) and (8) last week, choose 5 and determine whether they can be integrated and/or differentiated. (5, \#10)
	\item Let $f:[0,1] \to \R$ be continuous and one-to-one.  Show that $f$ is either increasing or decreasing. (5, \#25)
	\item Find functions $f, g:[0,1]\to \R$ such that 
	$$2 \|f\|^2 + 2\|g\|^2 \neq \|f+g\|^2 + \|f-g\|^2$$
	where $\|h\| = \sup\{|h(x)| \mid x \in [0,1]\}$ is the sup norm.  This means that the parallelogram law fails and that the sup norm does not come from an inner product (5, \#48, also see Chapter 1.7 and 1, \#12)
	\item Let $f_n:[0,1] \to \R$ be monotone increasing and continuous such that, for every $x \in [0,1]$, $F(x) = \sum_{n=0}^\infty f_n(x)$ converges (so the sequence of functions converges pointwise).  Show that $F$ is continuous. (5, \#69)
	\item We know that continuous functions are integrable, but we prove something more: Let $f:[a, b]\to \R$ be monotone (either increasing or decreasing).  Show that it is integrable.
	
\end{enumerate}



{\bf Niche problems:}
\begin{enumerate}
		\item We've shown that $exp_n$ converges absolutely and uniformly to $exp$ on compact sets.  Show that 
		\begin{itemize}
			\item $exp_n$ converges pointwise to $exp$ on $\R$.
			\item $exp_n$ \emph{does not} converge uniformly to $exp$ on $\R$.
		\end{itemize}
		Why is this last part not a problem in developing the properties of $exp$?
		
		\item Define $g:\R \to \R$ by setting $g(x) = |x|$ for $x \in [-\frac{1}{2}, \frac{1}{2}]$ and saying that $g$ has period $1$.  Define
		$$f(x) =\sum_{n=1}^\infty \frac{g(4^{n-1}x)}{4^{n-1}}$$
		\begin{enumerate}
			\item Sketch $g$ and the first few terms of the sum.
			\item Use the Weierstrass $M$ test to show that $f$ is continuous.
			\item Prove that $f$ is differentiable at no point.
		\end{enumerate}
		
		\item There are lots of neat results that I don't think we'll have time for.  One of them is Taylor's theorem, so it appears here as a Niche problem.  If I do get to it, lucky for you!
		
		Let $f:\R \to \R$ have a $(k+1)$th derivative \underline{on all of $\R$}.  For each $x \in R$, there is a number $\alpha$ between $a$ and $x$ such that
		$$f(x) = f(a) + f'(a) (x-a) + \frac{f^{(2)}(a)}{2!}(x-a)^2 + \dots + \frac{f^{(k)}(a)}{k!}(x-a^k) + \frac{f^{(k+1)}(\alpha)}{(k+1)!}(x-a)^{n+1}$$
		
		\item Measure theory is a continuation of what we've learned that makes more functions integrable by extending the notion of volume to more sets in $\R^n$ (see Math 114).  For instance, while we know the function
		$$f(x) = \begin{cases} 1 & x \in \Q\\ 0 & x \not\in \Q\end{cases}$$
		is not integrable via the (Riemann) integral we have defined, it is integrable using the Lebesgue integral.
		
		Developing this notion of measure or volume is hard, but the definition of having volume 0 is easier.
		\begin{definition}
		Let $A \subset \R$.  We say that $A$ \emph{has measure zero} iff, for every $\epsilon>0$, there are open intervals $(a_n, b_n) \subset R$ for $n \in \N$ such that
		\begin{enumerate}
			\item $A \subset \cup_{n=0}^\infty (a_n, b_n)$ and
			\item $\sum_{n=0}^\infty (b_n-a_n) < \epsilon$.
		\end{enumerate}
		\end{definition}
		
		Show that $\Q$ (the rationals) and $C$ (the Cantor set) both have measure 0.
		
		\item The following exercise is also useful for measure theory: A function $g:[0,1]\to \R$ is called \emph{simple} if we can divide $[0,1]$ up into subintervals on which $g$ is constant, except perhaps at the endpoints.  Let $f:[0,1]\to \R$ be continuous and $\epsilon >0$.  Show that there is a simple function $g$ such that
		$$\|f-g\| = \sup\{|f(x) - g(x)|\mid x\in [0,1]\} < \epsilon$$
		(5, \#39)
\end{enumerate}

{\bf Doc Brown problems:}
\begin{enumerate}
	\item Study for your midterms.
	\item Live a happy and fulfilling life post-Math 112.
\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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