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\begin{document}
\iftitle{Analysis Problems\\{\normalsize Integration Workshop 2019, 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 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.
\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.)}
\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 is called {\em connected} if it cannot be represented as a union of disjoint nonempty 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}