DZG: this comes from some tex file that I found when studying for quals, so is definitely not my own content! I’ve just copied it here for extra practice.

May 2016 Qual

May 2016, 1

Consider the function $f(x) = \frac{x}{1-x^2}$, $x \in (0,1)$.

• By using the $\epsilon-\delta$ definition of the limit only, prove that $f$ is continuous on $(0,1)$. (Note: You are not allowed to trivialize the problem by using properties of limits).

• Is $f$ uniformly continuous on $(0,1)$? Justify your answer.

(May 2016, 2)

Let $\{a_k\}_{k=1}^\infty$ be a bounded sequence of real numbers and $E$ given by: $$\begin{align*}E:= \bigg\{s \in \mathbb{R}\, \colon \, \text{ the set } \{k \in \mathbb{N}\, \colon \, a_k \geq s\} \text{ has at most finitely many elements}\bigg\}.\end{align*}$$ Prove that $\limsup_{k \to \infty} a_k = \inf E$.

(May 2016, 3)

Assume $(X,d)$ is a compact metric space.

• Prove that $X$ is both complete and separable.

• Suppose $\{x_k\}_{k=1}^\infty \subseteq X$ is a sequence such that the series $\sum_{k=1}^\infty d(x_k, x_{k+1})$ converges. Prove that the sequence $\{x_k\}_{k=1}^\infty$ converges in $X$.

(May 2016, 4)

Suppose that $f \colon [0,2] \to \mathbb{R}$ is continuous on $[0,2]$ , differentiable on $(0,2)$, and such that $f(0) = f(2) = 0$, $f(c) = 1$ for some $c \in (0,2)$. Prove that there exists $x \in (0,2)$ such that $|f'(x)| >1.$

(May 2016, 5)

Let $f_n(x) = n^\beta x(1-x^2)^n$, $x \in [0,1]$, $n \in \mathbb{N}$.

• Prove that $\{f_n\}_{n=1}^\infty$ converges pointwise on $[0,1]$ for every $\beta \in \mathbb{R}$.

• Show that the convergence in part (a) is uniform for all $\beta < \frac{1}{2}$, but not uniform for any $\beta \geq \frac{1}{2}$.

(May 2016, 6)

• Suppose $f \colon [-1,1] \to \mathbb{R}$ is a bounded function that is continuous at $0$. Let $\alpha(x) = -1$ for $x \in [-1,0]$ and $\alpha(x)=1$ for $x \in (0,1]$. Prove that $f \in \mathcal{R}(\alpha)[-1,1]$, i.e., $f$ is Riemann integrable with respect to $\alpha$ on $[-1,1]$, and $\int_{-1}^1 f d\alpha = 2f(0)$.

• Let $g \colon [0,1] \to \mathbb{R}$ be a continuous function such that $\int_0^1 g(x)x^{3k+2} dx = 0$ for all $k = 0, 1, 2, \ldots$. Prove that $g(x) =0$ for all $x \in [0,1]$.

Metric Spaces and Topology

(May 2019, 1)

Let $(M, d_M)$, $(N, d_N)$ be metric spaces. Define $d_{M \times N} \colon (M \times N) \times (M \times N) \to \mathbb{R}$ by $$\begin{align*}d_{M \times N}((x_1, y_1), (x_2, y_2)) := d_M(x_1, x_2) + d_N(y_1, y_2).\end{align*}$$

• Prove that $(M \times N, d_{M \times N})$ is a metric space.

• Let $S \subseteq M$ and $T \subseteq N$ be compact sets in $(M, d_M)$ and $(N, d_N)$, respectively. Prove that $S \times T$ is a compact set in $(M \times N, d_{M \times N})$.

(June 2003, 1b,c)

• Show by example that the union of infinitely many compact subsets of a metric space need not be compact. (c) If $(X,d)$ is a metric space and $K\subset X$ is compact, define $d(x_0,K)=\inf_{y\in K} d(x_0,y)$. Prove that there exists a point $y_0\in K$ such that $d(x_0,K)=d(x_0,y_0)$.

(January 2009, 4a)

Consider the metric space $(\mathbb{Q},d)$ where $\mathbb{Q}$ denotes the rational numbers and $d(x,y)=|x-y|$. Let $E=\{x\in\mathbb{Q}:x>0,\,2<x^2<3\}$. Is $E$ closed and bounded in $\mathbb{Q}$? Is $E$ compact in $\mathbb{Q}$?

(January 2011 3a)

Let $(X,d)$ be a metric space, $K\subset X$ be compact, and $F\subset X$ be closed. If $K\cap F=\emptyset$, prove that there exists an $\epsilon>0$ so that $d(k,f)\geq \epsilon$ for all $k\in K$ and $f\in F$.

5?

Let $(X,d)$ be an unbounded and connected metric space. Prove that for each $x_0 \in X$, the set $\{x \in X \, \colon \, d(x,x_0) = r\}$ is nonempty.

Sequences and Series

(June 2013 1a)

Let $a_n =\sqrt{n}\left(\sqrt{n+1}-\sqrt{n}\right)$. Prove that $\lim_{n\to\infty}a_n=1/2$.

(January 2014 2)

• Produce sequences $\{a_n\},\,\{b_n\}$ of positive real numbers such that $$\begin{align*}\liminf_{n\to\infty}(a_nb_n)>\left(\liminf_{n\to\infty} a_n\right) \left(\liminf_{n\to\infty} b_n\right).\end{align*}$$

• If $\{a_n\},\,\{b_n\}$ are sequences of positive real numbers and $\{a_n\}$ converges, prove that $$\begin{align*}\liminf_{n\to\infty}(a_nb_n)=\left(\lim_{n\to\infty}a_n\right)\left(\liminf_{n\to\infty}b_n\right).\end{align*}$$

(May 2011 4a)

Determine the values of $x\in\mathbb{R}$ for which $\displaystyle\sum_{n=1}^\infty \frac{x^n}{1+n|x|^n}$ converges, justifying your answer carefully.

(June 2005 3b)

If the series $\sum_{n=0}^\infty a_n$ converges conditionally, show that the radius of convergence of the power series $\sum_{n=0}^\infty a_nx^n$ is 1.

(January 2011 5)

Suppose $\{a_n\}$ is a sequence of positive real numbers such that $\lim_{n\to\infty}a_n=0$ and $\sum a_n$ diverges. Prove that for all $x>0$ there exist integers $n(1)<n(2)<\ldots$ such that $\sum_{k=1}^\infty a_{n(k)}=x$.\

(Note: Many variations on this problem are possible including more general rearrangements. You may also wish to show that if $\sum a_n$ converges conditionally then given any $x\in\mathbb{R}$ there is a rearrangement of $\{b_n\}$ of $\{a_n\}$ such that $\sum b_n=r$. See Rudin Thm. 3.54 for a further generalization.)

(June 2008 # 4b)

Assume $\beta >0$, $a_n>0$, $n=1,2,\ldots$, and the series $\sum a_n$ is divergent. Show that $\displaystyle \sum \frac{a_n}{\beta + a_n}$ is also divergent.

Continuity of Functions

Differential Calculus

(June 2005 1a)

Use the definition of the derivative to prove that if $f$ and $g$ are differentiable at $x$, then $fg$ is differentiable at $x$.

(January 2006 2b)

Assume that $f$ is differentiable at $a$. Evaluate $$\begin{align*}\lim_{x\to a}\frac{a^nf(x)-x^nf(a)}{x-a},\quad n\in\mathbb{N}.\end{align*}$$

(June 2007 3a)

Suppose that $f,g:\mathbb{R}\to\mathbb{R}$ are differentiable, that $f(x)\leq g(x)$ for all $x\in\mathbb{R}$, and that $f(x_0)=g(x_0)$ for some $x_0$. Prove that $f'(x_0)=g'(x_0)$.

(June 2008 3a)

Prove that if $f'$ exists and is bounded on $(a,b]$, then $\lim_{x\to a^+}f(x)$ exists.

(January 2012 4b, extended)

Let $f:\mathbb{R}\to\mathbb{R}$ be a differentiable function with $f'\in C(\mathbb{R})$. Assume that there are $a,b\in\mathbb{R}$ with $\lim_{x\to\infty}f(x)=a$ and $\lim_{x\to\infty}f'(x)=b$. Prove that $b=0$. Then, find a counterexample to show that the assumption $\lim_{x\to\infty}f'(x)$ exists is necessary.

(June 2012 1a)

Suppose that $f:\mathbb{R}\to\mathbb{R}$ satisfies $f(0)=0$. Prove that $f$ is differentiable at $x=0$ if and only if there is a function $g:\mathbb{R}\to\mathbb{R}$ which is continuous at $x=0$ and satisfies $f(x)=xg(x)$ for all $x\in\mathbb{R}$.

Integral Calculus

(January 2006 4b)

Suppose that $f$ is continuous and $f(x)\geq 0$ on $[0,1]$. If $f(0)>0$, prove that $\int_0^1 f(x)dx>0$.

(June 2005 1b)

Use the definition of the Riemann integral to prove that if $f$ is bounded on $[a,b]$ and is continuous everywhere except for finitely many points in $(a,b)$, then $f\in\mathscr{R}$ on $[a,b]$.

(January 2010 5)

Suppose that $f:[a,b]\to\mathbb{R}$ is continuous, $f\geq 0$ on $[a,b]$, and put $M=\sup\{f(x):x\in[a,b]\}$. Prove that $$\begin{align*}\lim_{p\to\infty}\left(\int_a^b f(x)^p\,dx\right)^{1/p}=M.\end{align*}$$

(January 2009 4b)

Let $f$ be a continuous real-valued function on $[0,1]$. Prove that there exists at least one point $\xi\in[0,1]$ such that $\int_0^1 x^4 f(x)\,dx=\frac{1}{5}f(\xi)$.

(June 2009 5b)

Let $\phi$ be a real-valued function defined on $[0,1]$ such that $\phi$, $\phi'$, and $\phi''$ are continuous on $[0,1]$. Prove that $$\begin{align*}\int_0^1 \cos x \frac{x\phi'(x)-\phi(x)+\phi(0)}{x^2}\,dx<\frac{3}{2}||\phi''||_\infty,\end{align*}$$ where $||\phi''||_\infty = \sup_{[0,1]}|\phi''(x)|.$ Note that $3/2$ may not be the smallest possible constant.

Sequences and Series of Functions

(June 2010 6a)

Let $f:[0,1]\to\mathbb{R}$ be continuous with $f(0)\neq f(1)$ and define $f_n(x)=f(x^n)$. Prove that $f_n$ does not converge uniformly on $[0,1]$.

(January 2008 5a)

Let $f_n(x) = \frac{x}{1+nx^2}$ for $n \in \mathbb{N}$. Let $\mathcal{F} := \{f_n \, \colon \, n = 1, 2, 3, \ldots\}$ and $[a,b]$ be any compact subset of $\mathbb{R}$. Is $\mathcal{F}$ equicontinuous? Justify your answer.

(January 2005 4, June 2010 6b)

If $f:[0,1]\to\mathbb{R}$ is continuous, prove that $$\begin{align*}\displaystyle\lim_{n\to\infty}\int_0^1 f(x^n)\,dx=f(0).\end{align*}$$

(January 2020 4a)

Let $M<\infty$ and $\mathcal{F} \subseteq C[a,b]$. Assume that each $f \in \mathcal{F}$ is differentiable on $(a,b)$ and satisfies $|f(a)| \leq M$ and $|f'(x)| \leq M$ for all $x \in (a,b)$. Prove that $\mathcal{F}$ is equicontinuous on $[a,b]$.

(June 2005 5)

Suppose that $f\in C([0,1])$ and that $\displaystyle \int_0^1 f(x)x^n\,dx=0$ for all $n=99,100,101,\ldots$. Show that $f\equiv 0$.\

Note: Many variations on this problem exist. See June 2012 6b and others.

(January 2005 3b)

Suppose $f_n:[0,1]\to\mathbb{R}$ are continuous functions converging uniformly to $f:[0,1]\to\mathbb{R}$. Either prove that $\displaystyle\lim_{n\to\infty}\int_{1/n}^1 f_n(x)\,dx=\int_0^1 f(x)\,dx$ or give a counterexample.

Miscellaneous Topics

Bounded Variation

(January 2018)

Let $f \colon [a,b] \to \mathbb{R}$. Suppose $f \in \text{BV}[a,b]$. Prove $f$ is the difference of two increasing functions.

(January 2007, 6a)

Let $f$ be a function of bounded variation on $[a,b]$. Furthermore, assume that for some $c>0$, $|f(x)| \geq c$ on $[a,b]$. Show that $g(x) = 1/f(x)$ is of bounded variation on $[a,b]$.

(January 2017, 2a)

Define $f \colon [0,1] \to [-1,1]$ by $$\begin{align*}f(x):= \begin{cases} x\sin\big({\frac{1}{x}}\big) & 0 < x \leq 1 \\ 0 & x = 0 \end{cases}\end{align*}$$ Determine, with justification, whether $f$ is if bounded variation on the interval $[0,1]$.

(January 2020, 6a)

Let $\{a_n\}_{n=1}^\infty \subseteq \mathbb{R}$ and a strictly increasing sequence $\{x_n\}_{n=1}^\infty \subseteq (0,1)$ be given. Assume that $\sum_{n=1}^\infty a_n$ is absolutely convergent, and define $\alpha \colon [0,1] \to \mathbb{R}$ by $$\begin{align*}\alpha(x):= \begin{cases} a_n & x=x_n \\ 0 & \text{otherwise} \end{cases}.\end{align*}$$ Prove or disprove: $\alpha$ has bounded variation on $[0,1]$.

Metric Spaces and Topology

• Find an example of a metric space $X$ and a subset $E \subseteq X$ such that $E$ is closed and bounded but not compact.

(May 2017 6)

Let $(X,d)$ be a metric space. A function $f \colon X \to \mathbb{R}$ is said to be lower semi-continuous (l.s.c) if $f^{-1}(a,\infty) = \{x \in X \, \colon \, f(x)> a\}$ is open in $X$ for every $a \in \mathbb{R}$. Analogously, $f$ is upper semi-continuous (u.s.c) if $f^{-1}(-\infty, b) = \{x \in X \, \colon \, f(x)<b\}$ is open in $X$ for every $b \in \mathbb{R}$.

• Prove that a function $f \colon X \to \mathbb{R}$ is continuous if and only if $f$ is both l.s.c. and u.s.c.

• Prove that $f$ is lower semi-continuous if and only if $\liminf_{n \to \infty} f(x_n) \geq f(x)$ whenever $\{x_n\}_{n=1}^\infty \subseteq X$ such that $x_n \to x$ in $X$.

(January 2017 3)

Let $(X,d)$ be a compact metric space. Suppose that $f_n \colon X \to [0,\infty)$ is a sequence of continuous functions with $f_n(x) \geq f_{n+1}(x)$ for all $n \in \mathbb{N}$ and $x \in X$, and such that $f_n \to 0$ pointwise on $X$. Prove that $\{f_n\}_{n=1}^\infty$ converges uniformly on $X$.

Integral Calculus

1.

(June 2014 1)Define $\alpha \colon [-1,1] \to \mathbb{R}$ by $$\begin{align*}\alpha(x) := \begin{cases} -1 & x \in [-1,0] \\ 1 & x \in (0,1]. \end{cases}\end{align*}$$ Let $f \colon [-1,1] \to \mathbb{R}$ be a function that is uniformly bounded on $[-1,1]$ and continuous at $x=0$, but not necessarily continuous for $x \neq 0$. Prove that $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ over $[-1,1]$ and that $$\begin{align*}\int_{-1}^1 f(x)d\alpha(x) = 2f(0).\end{align*}$$

(June 2017 2)

Prove : $f \in \mathcal{R}(\alpha)$ on $[a,b]$ if and only if for any $a <c<b$, $f \in \mathcal{R}(\alpha)$ on $[a,c]$ and on $[c,b]$. In addition, if either condition holds, then we have that $$\begin{align*}\int_a^c fd\alpha + \int_c^b fd\alpha = \int_a^b fd\alpha.\end{align*}$$

(Spring 2017 7)

Prove that if $f \in \mathcal{R}$ on $[a,b]$ and $\alpha \in C^1[a,b]$, then the Riemann integral $\int_a^b f(x)\alpha'(x)dx$ exists and $$\begin{align*}\int_a^b f(x) d\alpha(x)= \int_a^b f(x)\alpha'(x)dx.\end{align*}$$

Sequences and Series (and of Functions)

(January 2006 1)

Let the power series series $\sum_{n=0}^\infty a_nx^n$ and $\sum_{n=0}^\infty b_nx^n$ have radii of convergence $R_1$ and $R_2$, respectively.

?

If $R_1 \neq R_2$, prove that the radius of convergence, $R$, of the power series $\sum_{n=0}^\infty (a_n+b_n)x^n$ is $\min\{R_1, R_2\}$. What can be said about $R$ when $R_1 = R_2$?

?

Prove that the radius of convergence, $R$, of $\sum_{n=0}^\infty a_nb_nx^n$ satisfies $R \geq R_1R_2$. Show by means of example that this inequality can be strict.

?

Show that the infinite series $\sum_{n=0}^\infty x^n2^{-nx}$ converges uniformly on $[0,B]$ for any $B > 0$. Does this series converge uniformly on $[0,\infty)$?

(January 2006 4a)

Let $$\begin{align*}f_n(x) = \begin{cases} \frac{1}{n} & x \in (\frac{1}{2^{n+1}}, \frac{1}{2^n}] \\ 0 & \text{ otherwise}.\end{cases}\end{align*}$$

Show that $\sum_{n=1}^\infty f_n$ does not satisfy the Weierstrass M-test but that it nevertheless converges uniformly on $\mathbb{R}$.

?

Let $f_n \colon [0,1) \to \mathbb{R}$ be the function defined by $$\begin{align*}f_n(x):= \sum_{k=1}^n \frac{x^k}{1+x^k}.\end{align*}$$

• Prove that $f_n$ converges to a function $f \colon [0,1) \to \mathbb{R}$.

• Prove that for every $0 < a < 1$ the convergence is uniform on $[0,a]$.

• Prove that $f$ is differentiable on $(0,1)$.

January 2019 Qualifying Exam

• Suppose that $f: [0,1] \to \mathbb{R}$ is differentiable and $f(0) = 0$. Assume that there is a $k > 0$ such that $$\begin{align*}|f'(x)| \leq k|f(x)|\end{align*}$$ for all $x\in [0,1]$. Prove that $f(x) = 0$ for all $x\in [0,1]$.