\documentclass{article}

\usepackage{fullpage,palatino,mathpazo,titlesec,MnSymbol}


\newcommand{\iftitle}[1]{\addtolength{\baselineskip}{2ex}{\par\noindent{\fontfamily{put}\LARGE\bfseries{#1}}}\par\vspace{4ex}\addtolength{\baselineskip}{-2ex}}
\newcommand{\ifauthor}[2]{{\fontfamily{put}\noindent\rule{0pt}{3.5ex}{#1}\\{\small\it{#2}}\\}}


\newcounter{exercise}
\newcommand{\prb}{\stepcounter{exercise}\par\vspace{1ex}\noindent{\bfseries\quad\arabic{exercise}.}\quad}

\newenvironment{prob}[1]{\par\vspace{2.5ex}\noindent\large\sc\bfseries\hfill\underline{#1}\par\rm\vspace{2ex}\normalsize}{\par\vspace{4ex}}




\newcommand{\spc}[1]{{\sf{{#1}}}}



\newcommand{\R}{{\mathbb R}}
\DeclareMathOperator{\me}{{\mathop e}}
\newcommand{\set}[1]{{\mathcal{#1}}}
\DeclareMathOperator{\D}{{\boldsymbol{\mathrm D\!}}}
\newcommand{\md}{\mathop{\mathrm d\!}}
\DeclareMathOperator{\pd}{{\partial}}







\begin{document}






\iftitle{Analysis Problems\\{\normalsize Integration Workshop 2017, Department of Mathematics, University of Arizona}}
\vspace{-3ex}\ifauthor{Compiled by Ibrahim Fatkullin}{ibrahim@math.arizona.edu}
\vspace{0.25in}





\begin{prob}{Basic concepts, sequences, convergence}

	\prb Every bounded increasing sequence in $\R$ is Cauchy (do not use completeness of $\R$). Every bounded sequence in $\R^n$ has a converging subsequence. 
	
	\prb A metric space is compact if and only if it is complete and totally bounded, i.e.,  for every $r>0$, it may be covered by a finite number of open balls of radius $r$.

	\prb Every open set in $\R$ is a union of at most a countable number of disjoint open intervals.
	
	\prb Compute $\displaystyle\lim_{n\to\infty}\sqrt[n]{n}$ without explicitly using that $(\ln n)/n\to0$ as $n\to\infty$.

\end{prob}

\begin{prob}{Series}

	\prb Prove that $\me$ is irrational. {\em (Hint: estimate approximation errors for partial sums of a series representation of $\me$ or some appropriate quantity related to it.)}

	\prb A conditionally convergent series (i.e., a non-absolutely convergent series whose partial sums still converge) may be summed to any desired number by an appropriate {\em rearrangement} of its terms.

	\prb A sequence is called {\em Ces\`aro summable} if the arithmetic means of its partial sums converge. What is the value of the sum $1-1+1-1+\cdots$ in Ces\`aro sense?

	\prb Find examples of converging and diverging  series for which $\displaystyle\lim_{n\to\infty}|x_{n+1}/x_n|=1$ and $\displaystyle\lim_{n\to\infty}\sqrt[n]{|x_n|}=1$.

	\prb If the coefficients of a power series are integers, infinitely many of which are nonzero (i.e., the series is not a polynomial) then the radius of convergence of this series is at most 1.
	
	\prb Prove that the radii of convergence of power series $\displaystyle\sum_{n=0}^\infty a_nx^n$ and $\displaystyle\sum_{n=1}^\infty na_nx^{n-1}$ are the same.

	\prb Suppose all $x_n\geq 0$ and the series $\displaystyle \sum_{n=0}^\infty x_n$ converges. Set $\displaystyle y_n=\sum_{m=n}^\infty x_m$. Prove that $\displaystyle\sum_{n=0}^\infty x_n/y_n$ diverges, while $\displaystyle\sum_{n=0}^\infty x_n/\sqrt{y_n}$ converges.
	
	\prb Prove that $\displaystyle\sum_{n=1}^\infty x_n$ converges iff $\displaystyle\prod_{n=1}^\infty (1+x_n)$ converges.

\end{prob}


\newpage
\begin{prob}{Continuity}
	
	\prb Prove that a map of a metric space into a metric space is continuous iff pre-image of every open set is open. Show that if we replaced ``pre-image'' by ``image,'' the statement would be wrong.
	
	\prb An image of a compact set under a continuous map is compact.
		
	\prb {\bf Intermediate value theorem.} An image of a connected set under a continuous map is connected. (A set $\set A$ is called {\em connected} if it cannot be {\em minimally} covered by two disjoint open sets.)
	
	\prb Level sets of continuous functions are closed.
		
	\prb A function continuous on a compact set $\set A$ is also uniformly continuous on $\set A$.
	
	\prb A function $f:\set X\to\R$ is called convex if  inequality, $$f\big(\alpha x_1+(1-\alpha)x_2\big)\leq\alpha f(x_1)+(1-\alpha)f(x_2)$$ holds for all $x_1,x_2\in\set X$ and $\alpha\in[0,1]$. Prove that convex functions are continuous.
	
	\prb  A function $f:\set X\to\set Y$ is called {\bf H\"older continuous} with exponent $\alpha\in[0,1]$ if there exists some constant $C$ such that for all $x_1,x_2\in\set X$, $$ d_{\set Y}\big(f(x_1),f(x_2)\big)\leq Cd_{\!\set X}^{\,\alpha}(x_1,x_2).$$ Prove that if $\alpha>0$, $f$ is continuous; if $\alpha>1$, $f$ is constant. (H\"older continuity with $\alpha=1$ is also referred to as {\bf Lipschitz} continuity.)
	
	\prb Construct a function $f:\R\to\R$ which is continuous on all irrationals and discontinuous on all rationals. Prove that the opposite is impossible.
\end{prob}


\begin{prob}{Differentiability}

	\prb Is there a function, differentiable on all irrationals and discontinuous on all rationals?
	
	\prb Suppose some {\em sublevel set,} $\set F=\{x: f(x)\leq F\}$, of a differentiable function $f:\set X\to\R$ is compact, then $f$ achieves its minimum at some $x\in\set F$, and its derivative at $x$ vanishes.
	
	\prb Prove Taylor's theorem using the mean value theorem.
			
	\prb If partial derivatives of $f:\R^n\to\R$ are bounded in a neighborhood of $x$, then $f$ is continuous at $x$.

	\prb Find a function discontinuous at the origin whose partial derivatives at the origin are nevertheless well-defined.

	\prb If there exists a function $\D f(x_0):\set X\to\set Y$, such that for all $x\in\set X$, $$\lim_{\epsilon\to0}\frac{\|f(x_0+\epsilon x)-f(x_0)-\epsilon \D f(x_0;x)\|}{\epsilon}=0,$$ it is called the {\bf directional (G\^ateaux) derivative} of $f$ at $x_0$. Give examples of  non-differentiable functions which are G\^ateaux-differentiable. {\em (Hint: this may happen if, e.g., $\D f(x_0;x)$ is not a linear map of $x$.)} Suppose $\D f(x_0)$ exists and is linear, would this imply Fr\'echet differentiability as well?
	
	\prb Give example of a function whose derivative at 0 is equal to 1, though the function itself is not invertible in any neighborhood of 0. 

	
\end{prob}



\newpage
\begin{prob}{Integration}

	\prb A function is of bounded variation iff it may be represented as a difference of two monotone-increasing functions.
	
	\prb {\bf Integral test for convergence of series.} Suppose $f:\R^+\to\R^+$ is monotone-decreasing, then $$\sum_{n=1}^\infty f(n)\quad\text{converges iff }\quad\int_1^\infty f(x)\md x\quad\text{converges.}$$
	
	\prb Prove that if $\displaystyle\int_0^1f(x)x^n\md x=0$  for all $n=0,1,2\ldots$ and $f$ is continuous, then $f\equiv 0$ on $[0,1]$.
	
	\prb Show by direct computation that $$\int_1^\infty\left(\int_1^\infty \frac{x^2-y^2}{(x^2+y^2)^2}\;\md y\right)\md x=-\int_1^\infty\left(\int_1^\infty\frac{x^2-y^2}{(x^2+y^2)^2}\;\md x\right)\md y=\frac{\pi}{4}.$$
	
	\prb Let $\Omega$ be an open bounded subset of $\R^2$ with smooth boundary $\pd\Omega$. Prove that $$\mathrm{Vol}(\Omega)=\iint_{\Omega}\md x\md y=\oint_{\pd\Omega} x\md y=-\oint_{\pd\Omega} y\md x=\frac{1}{2}\oint_{\pd\Omega} \big[x\md y-y\md x\big].$$
	
\end{prob}


\begin{prob}{Sequences of functions}
	
	\prb Partial sums of power series and their derivatives (of all orders) converge uniformly on compact subsets of their open intervals of convergence.
	
	\prb For real-valued functions on a metric space $\set X$, define the {\em supremum norm:} $$\|f\|=\sup_{x\in\set X}|f(x)|.$$ The set of all continuous functions for which $\|f\|<\infty$ is called $\spc C(\set X)$. When is $\spc C(\set X)$ a complete metric space with respect to the metric $d(f,g)=\|f-g\|$?

	\prb Suppose $\{f_n(x)\}$ is a sequence of differentiable functions converging uniformly to $f(x)$. Give an example illustrating that $f(x)$ need not be differentiable. Give an example illustrating that the derivatives $f^\prime_n(x)$ need not converge. Suppose that $f(x)$ is differentiable and $f^\prime_n(x)$ converge point-wise, show that the equality $\lim_{n\to\infty} f^\prime_n(x)=f^\prime(x)$ need not hold.
				
	\prb {\bf Peano's existence theorem.} Suppose $f:\R^2\to\R$ is continuous in a neighborhood of $(x_0,y_0).$ Then there exists a function $y(x)$, such that $y(x_0)=y_0$ and $y^\prime(x)=f\big(x,y(x)\big)$. {\em (Hint: construct Euler approximations to the solution of this differential equation and show that they constitute an equicontinuous family of functions.)}
	


\end{prob}














\end{document}