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Mathematical Analysis

Sam Macdonald

Appendix A Qualifying Exams

Winter 2023.

Suppose that \(f: \mathbb{R} \rightarrow \mathbb{R}\) is differentiable at \(x_{0} \in \mathbb{R}\text{.}\) Let \(\left\{a_{n}\right\}_{n=1}^{\infty} \subseteq\left(-\infty, x_{0}\right)\) and \(\left\{b_{n}\right\}_{n=1}^{\infty} \subseteq\left(x_{0}, \infty\right)\) be sequences that both converge to \(x_{0}\text{.}\) Prove that
\begin{equation*} \lim _{n \rightarrow \infty} \frac{f\left(b_{n}\right)-f\left(a_{n}\right)}{b_{n}-a_{n}}=f^{\prime}\left(x_{0}\right). \end{equation*}
(You may assume that \(x_{0}=0\) ).
Solution.
Coming (not so) soon!
Let \(\left\{a_{n}\right\}_{n=1}^{\infty},\left\{b_{n}\right\}_{n=1}^{\infty} \subset(0, \infty)\) be given.
  1. Assume that \(\limsup _{n \rightarrow \infty}\left(a_{n} / b_{n}\right)<\infty\text{.}\) Prove there exists \(M \in \mathbb{R}\) such that \(a_{n} \leq M b_{n}\text{,}\) for all \(n \in \mathbb{N}\text{.}\)
  2. Suppose the sequence \(\left\{\frac{a_{n}}{b_{n}}\right\}_{n=1}^{\infty}\) converges in \(\mathbb{R}\text{.}\) Must \(\limsup _{n \rightarrow \infty}\left(a_{n} / b_{n}\right)\) be finite?
Solution.
Coming (not so) soon!
  1. Suppose that \(\left\{f_{n}\right\}_{n=1}^{\infty}\) is a Cauchy sequence from \(\left(\mathcal{C}([0,1]), \rho_{\infty}\right)\text{.}\) Determine whether \(\left\{f_{n}\right\}_{n=1}^{\infty}\) must be uniformly equicontinuous.
  2. Suppose that \(\mathcal{F} \subseteq\left(\mathcal{C}([0,1]), \rho_{\infty}\right)\) is closed and bounded but not uniformly equicontinuous. Prove that \(\mathcal{F}\) is not compact in \(\left(\mathcal{C}([0,1]), \rho_{\infty}\right)\text{.}\)
Solution.
Coming (not so) soon!
Use the Riemann condition to show that \(f \in \mathcal{R}_{\alpha}[0,2]\) where \(f(x)=\tan (\pi x / 4)\) and
\begin{equation*} \alpha(x)=\begin{cases} x+1, \quad 0 \leq x \leq 1 \\ 4 x, \quad 1<x \leq 2 \end{cases}. \end{equation*}
Compute the value of the Riemann-Stieltjes integral \(\int_{0}^{2} f(x) d \alpha\text{.}\)
Solution.
Coming (not so) soon!
Determine all the values of \(x \in \mathbb{R}\) for which the series below converges
\begin{equation*} \sum_{n=1}^{\infty} \frac{x^{n}}{1+n|x|^{n}} \end{equation*}
Solution.
Coming (not so) soon!
Let \((M, d)\) be a metric space. Define the functions \(d^{\prime}, d^{\prime \prime}: M \times M \rightarrow[0, \infty)\) by
\begin{equation*} d^{\prime}(x, y)=\frac{d(x, y)}{1+d(x, y)}, \quad \text { and } \quad d^{\prime \prime}(x, y)=\min \{1, d(x, y)\}. \end{equation*}
  1. Show that \(d^{\prime}, d^{\prime \prime}\) are both metrics.
  2. Show that \(d, d^{\prime}, d^{\prime \prime}\) are equivalent metrics on the space \(M\text{.}\)
Solution.
Coming (not so) soon!

Spring 2022.

  1. Let \(f_{n}(x)=n x(1-x)^{n}\) for \(x \in[0,1]\text{.}\) Prove that \(\left\{f_{n}\right\}\) converges pointwise and determine if it converges uniformly on \([0,1]\text{.}\) Is \(\left\{f_{n}\right\}\) equicontinuous? Clearly motivate your answer.
  2. Prove that in general, if \(\left\{f_{n}\right\}_{n \geq 1}\) is an equicontinuous sequence of functions on a compact interval and \(f_{n} \rightarrow f\) pointwise, then the convergence is uniform.
Solution.
Coming (not so) soon!
Let \((X, \rho)\) be a metric space. Suppose that \(x_{0} \in X\text{.}\) For each \(\varepsilon>0\text{,}\) set
\begin{equation*} E_{\varepsilon}:=\left\{x \in X: \rho\left(x, x_{0}\right) \geq \varepsilon\right\}. \end{equation*}
Suppose that \(f: X \rightarrow \mathbb{R}\) is continuous and \(f\left(E_{\varepsilon}\right)\) is compact for all \(\varepsilon>0\text{.}\) Prove that \(f(X)\) is compact.
Solution.
Coming (not so) soon!
Consider the sequence \(\left\{x_{n}\right\}_{n \geq 1}\) defined by \(0<x_{1}<1\) and \(x_{n+1}=1-\sqrt{1-x_{n}}\) for \(n=\) \(1,2, \ldots\text{.}\) Show that \(x_{n} \rightarrow 0\) as \(n \rightarrow \infty\text{.}\) Also, show that \(\frac{x_{n+1}}{x_{n}} \rightarrow \frac{1}{2}\text{.}\)
Solution.
Coming (not so) soon!
Use the Riemann condition to show that \(f \in \mathcal{R}_{\alpha}[0,3]\) where \(f(x)=\ln (2 x+1)\) and
\begin{equation*} \alpha(x)=\begin{cases} x+2, & 0 \leq x \leq 2 \\ 3 x-1, & 2<x \leq 3 \end{cases} \end{equation*}
Compute the value of the Riemann-Stieltjes integral \(\int_{0}^{3} f(x) d \alpha\text{.}\)
Solution.
Coming (not so) soon!
Find the domain of convergence and the sum of the series
\begin{equation*} \sum_{n \geq 0}(-1)^{n} \frac{x^{2 n+1}}{2 n+1} \end{equation*}
Show how one may use the sum of the series to provide an approximation for \(\pi\) up to three decimals. Be sure to provide all technical details.
Solution.
Coming (not so) soon!
Show that
\begin{equation*} d_{1}(f, g):=\int_{0}^{1}|f(x)-g(x)| d x \quad \text { and } \quad d_{\infty}(f, g):=\operatorname{lub}_{x \in[0,1]}|f(x)-g(x)| \end{equation*}
are metrics on \(C[0,1]\text{,}\) but are not equivalent on \(C[0,1]\text{.}\)
Solution.
Coming (not so) soon!

Winter 2022.

Suppose \(a, b \in \mathbb{R}\) with \(a<b\) and let \(f:[a, b] \rightarrow \mathbb{R}\) be a differentiable function such that \(f^{\prime}:[a, b] \rightarrow \mathbb{R}\) is continuous. Show that for every \(\epsilon>0\text{,}\) there is a \(\delta>0\) such that for every \(x, y \in[a, b]\) with \(|x-y|<\delta\text{,}\) we have
\begin{equation*} \left|\frac{f(x)-f(y)}{x-y}-f^{\prime}(x)\right|<\epsilon. \end{equation*}
Solution.
Coming (not so) soon!
Let \(X\) and \(Y\) be metric spaces and let \(f: X \rightarrow Y\) be a continuous bijection.
  1. Show by example that \(f^{-1}: Y \rightarrow X\) need not be continuous.
  2. Show that if \(X\) is compact, then \(f^{-1}: Y \rightarrow X\) is continuous.
Solution.
Coming (not so) soon!
Compute, with proof, \(\lim _{k \rightarrow \infty} \sum_{n=1}^{\infty} n^{-k}\text{.}\)
Solution.
Coming (not so) soon!
  1. Let \(f:[1,2] \rightarrow \mathbb{R}\) be a continuous function. If \(\int_{1}^{2} x^{-n} f(x) d x=0\) for all integers \(n \geq 0\text{,}\) show that \(f=0\text{.}\)
  2. Let \(g:[1,2] \rightarrow \mathbb{R}\) be a differentiable function such that \(g^{\prime}:[1,2] \rightarrow \mathbb{R}\) is continuous. If \(\int_{1}^{2} x^{-n} d g(x)=0\) for all integers \(n \geq 0\text{,}\) show that \(g\) is constant.
Solution.
Coming (not so) soon!
Prove the following special case of Dini's Theorem: if \(\left(f_{n}:[0,1] \rightarrow \mathbb{R}\right)_{n=1}^{\infty}\) is decreasing sequence of continuous functions such that \(\lim _{n \rightarrow \infty} f_{n}(x)=0\) for all \(x \in[0,1]\text{,}\) then \(\left(f_{n}\right)_{n=1}^{\infty}\) converges uniformly to 0. (You should not use any form of Dini's theorem without proof.)
Solution.
Coming (not so) soon!
Let \(f:[0,1] \rightarrow \mathbb{R}\) be a continuous function.
  1. Show \(\lim _{n \rightarrow \infty} \int_{0}^{1} f\left(x^{1 / n}\right) d x=f(1)\text{.}\)
  2. If \(f(x)>0\) for all \(x \in[0,1]\text{,}\) show \(\lim _{n \rightarrow \infty} \int_{0}^{1} f(x)^{1 / n} d x=1\text{.}\)
Solution.
Coming (not so) soon!

Spring 2021.

Suppose that \(X\) is a compact metric space, \(\left(x_{n}\right)_{n=1}^{\infty} \subseteq X, x \in X\text{,}\) and every convergent subsequence of \(\left(x_{n}\right)_{n=1}^{\infty}\) converges to \(x\text{.}\) Show that \(\left(x_{n}\right)_{n=1}^{\infty}\) converges to \(x\text{.}\)
Solution.
Coming (not so) soon!
  1. Show that if \(f:[a, b] \rightarrow \mathbb{R}\) is a differentiable function and \(f^{\prime}:[a, b] \rightarrow \mathbb{R}\) is bounded, then \(f\) has bounded variation.
  2. Construct, with proof, a continuous function \(g:[a, b] \rightarrow \mathbb{R}\) which does not have bounded variation.
Solution.
Coming (not so) soon!
Suppose that \(f: \mathbb{R} \rightarrow \mathbb{R}\) is a differentiable function such that there is no \(x \in \mathbb{R}\) with \(f(x)=f^{\prime}(x)=0\text{.}\) Show that if \(a, b \in \mathbb{R}\) with \(a<b\text{,}\) then \(\{x \in[a, b]: f(x)=0\}\) is a finite set.
Solution.
Coming (not so) soon!
Let \(X\) be a metric space. A function \(f: X \rightarrow \mathbb{R}\) is called lower semicontinuous if for every convergent sequence \(\left(x_{n}\right)_{n=1}^{\infty} \subseteq X\text{,}\) we have
\begin{equation*} f\left(\lim _{n \rightarrow \infty} x_{n}\right) \leq \liminf _{n \rightarrow \infty} f\left(x_{n}\right). \end{equation*}
  1. Let \(A \subseteq X\) be a set and define
    \begin{equation*} \chi_{A}: X \rightarrow \mathbb{R}: x \mapsto \begin{cases}1 & x \in A \\ 0 & x \notin A.\end{cases} \end{equation*}
    Show that \(\chi_{A}\) is lower semicontinuous if and only if \(A\) is open.
  2. Show that if \(X\) is a compact metric space and \(f: X \rightarrow \mathbb{R}\) is lower semicontinuous, then there is a \(c \in X\) such that
    \begin{equation*} f(c)=\inf \{f(x): x \in X\}. \end{equation*}
Solution.
Coming (not so) soon!
Suppose that \(\left(a_{n}\right)_{n=1}^{\infty}\) and \(\left(b_{n}\right)_{n=1}^{\infty}\) are sequences of strictly positive real numbers such that \(\sum_{n=1}^{\infty} b_{n}\) converges, and suppose that for each integer \(n \geq 1\text{,}\) we have \(\frac{a_{n+1}}{a_{n}} \leq \frac{b_{n+1}}{b_{n}}\text{.}\) Show that \(\sum_{n=1}^{\infty} a_{n}\) converges.
Solution.
Coming (not so) soon!
A function \(f:[0,1] \rightarrow \mathbb{R}\) is called Lipschitz if there is a constant \(C>0\) such that for all \(x, y \in[0,1]\text{,}\) we have
\begin{equation*} |f(x)-f(y)| \leq C|x-y| \end{equation*}
Let \(\operatorname{Lip}([0,1])\) denote the set of all Lipschitz functions \([0,1] \rightarrow \mathbb{R}\) with the uniform metric
\begin{equation*} d_{\infty}(f, g)=\sup \{|f(t)-g(t)|: 0 \leq t \leq 1\}, \quad f, g \in \operatorname{Lip}([0,1]). \end{equation*}
Show that \(\operatorname{Lip}([0,1])\) is a countable union of compact sets.
Solution.
Coming (not so) soon!

Winter 2021.

Suppose: \((X, d)\) and \((Y, \rho)\) are metric spaces; \((Y, \rho)\) is compact; and \(\phi: Y \rightarrow X\) is a continuous and onto function.
  1. A well-known theorem states that if \(F \subseteq Y\) is compact, then \(\phi(F)\) is also compact. Prove this theorem, and conclude that \((X, d)\) is a compact metric space.
  2. Suppose \(G \subseteq X\) and \(\phi^{-1}(G)\) is an open set. Prove that \(G\) is an open set.
Solution.
Coming (not so) soon!
Let \(\left(a_{n}\right)_{n=1}^{\infty}\) be a bounded sequence of real numbers. Being sure to include all details, prove that
\begin{equation*} \liminf _{n \rightarrow \infty} a_{n} \leq \liminf _{n \rightarrow \infty} \frac{a_{1}+a_{2}+\cdots+a_{n}}{n}. \end{equation*}
Hint.
Notice that for \(n, N \in \mathbb{N}\) and \(n>N, \frac{a_{1}+\cdots+a_{N}}{n}+\frac{a_{N+1}+\cdots+a_{n}}{n}=\frac{a_{1}+\cdots+a_{n}}{n}\text{.}\) Approximate \(\frac{a_{N+1}+\cdots+a_{n}}{n}\) in terms of \(\liminf _{n \rightarrow \infty} a_{n}\) and examine what happens if you hold \(N\) fixed and let \(n\) grow.
Solution.
Coming (not so) soon!
Suppose \(f:[a, b] \rightarrow \mathbb{R}\) is Riemann integrable and \(\int_{a}^{b}|f(t)| d t=0\text{.}\)
  1. If \(f\) is continuous, prove that \(f(t)=0\) for every \(t \in[a, b]\text{.}\)
  2. Give an example (with proof) of a non-zero Riemann integrable function such that \(\int_{a}^{b}|f(t)| d t=0\text{.}\)
Solution.
Coming (not so) soon!
Let \(Y \subseteq[0,1]\) be the usual Cantor \(\operatorname{set}^{1}\) and let \(C(Y)\) be the collection of all continuous complex-valued functions on \(Y\text{.}\)  1  A function \(p \in C(Y)\) is a projection if for every \(y \in Y, p(y)^{2}=p(y)\text{.}\) Given \(f \in C(Y)\) and \(\varepsilon>0\text{,}\) prove that there exists \(n \in \mathbb{N}\text{,}\) projections \(\left\{p_{k}\right\}_{k=1}^{n} \subseteq C(Y)\) and complex numbers \(\left\{\alpha_{k}\right\}_{k=1}^{n}\) such that for every \(y \in Y\text{,}\)
\begin{equation*} \left|f(y)-\sum_{k=1}^{n} \alpha_{k} p_{k}(y)\right|<\varepsilon. \end{equation*}
Solution.
Coming (not so) soon!
Suppose \(\left(a_{n}\right)\) is a decreasing sequence of real numbers and \(\sum_{n=1}^{\infty} a_{n}\) converges. Prove that \(\lim _{n \rightarrow \infty} n a_{n}=0\text{.}\)
Solution.
Coming (not so) soon!
For \(x \in \mathbb{R}\text{,}\) consider the series, \(\sum_{n=1}^{\infty} \frac{1}{n^{2}+x^{2}}\text{.}\)
  1. Prove this series converges for every \(x \in \mathbb{R}\text{.}\)
  2. Set \(f(x)=\sum_{n=1}^{\infty} \frac{1}{n^{2}+x^{2}}\text{.}\) Prove that \(f\) is differentiable at each \(x \in \mathbb{R}\text{.}\) Also, find a formula for \(f^{\prime}(x)\) (in terms of a series), being sure to justify that your formula is correct.
Solution.
Coming (not so) soon!

Spring 2020.

Suppose for each \(n \in \mathbb{N}, f_{n}:[0,1] \rightarrow \mathbb{R}\) is continuous and satisfies \(\sup \left\{\left|f_{n}(x)\right|: x \in[0,1]\right\} \leq 2020\text{.}\) Put \(g_{n}(x)=\int_{0}^{x} f_{n}(t) d t\text{.}\) Prove that there is a subsequence of \(\left(g_{n}\right)\) which converges uniformly on \([0,1]\text{.}\)
Solution.
It's enough to show that \(\overline{\{{g_n}\,:\,{n\geq 1}\}}\subseteq C[0,1]\) is compact. By the Arzelá-Ascoli Theorem, \(\{{g_n}\,:\,{n\geq 1}\}\) is bounded and equicontinuous.
Bounded: If \(n\geq 1,x\in[0,1]\text{,}\)
\begin{equation*} \begin{aligned} \left|{g_n(x)}\right| &=\left|{\int_0^x f_n(t)\,dt)}\right| \leq\int_0^x\left|{f_n(t)}\right|\,dt \leq 2020x \leq 2020. \end{aligned} \end{equation*}
Then for all \(n\geq 1\text{,}\) \(\left|{\left|{{g_n}}\right|}\right|_\infty\leq 2020\) so \(\{{g_n}\,:\,{n\geq 1}\}\) is bounded by 2020.
Equicontinuous: Fix \(\varepsilon>0\) and let \(\delta=\frac{\varepsilon}{2020}\text{.}\) Then if \(x,y\in[0,1]\) with \(\left|{x-y}\right|<\delta\) and \(n\geq 1\text{,}\)
\begin{equation*} \begin{aligned} \left|{g_n(x)-g_n(y)}\right| &=\left|{\int_0^x f_n(t)\,dt-\int_0^yf_n(t)\,dt}\right| =\left|{\int_y^x f_n(t)\,dt}\right| \leq \int_{\min\{{x,y}\}}^{\max\{{x,y}\}}\left|{f_n(t)}\right|\,dt\\ &\leq 2020\left|{x-y}\right| <\varepsilon \end{aligned} \end{equation*}
so \(\{{g_n}\,:\,{n\geq 1}\}\) is equicontinuous.
Suppose \(f: \mathbb{R} \rightarrow \mathbb{R}\) is twice differentiable and that \(f^{\prime \prime}(t)<0\) for every \(t \in \mathbb{R}\text{.}\) Prove that for any real numbers satisfying \(a<x<b\text{,}\)
\begin{equation*} (f(x)-f(a))(b-a)>(f(b)-f(a))(x-a). \end{equation*}
Solution.
Coming (not so) soon!
Suppose \((X, d)\) is a compact metric space.
  1. Let \(E\) be a closed subset of \(X\) and for \(x \in X\text{,}\) define \(\operatorname{dist}(x, E)=\inf \{d(x, e): e \in E\}\text{.}\) Show that if \(x \in X \backslash E\text{,}\) then \(\operatorname{dist}(x, E)>0\text{.}\)
  2. Suppose \(f: X \rightarrow X\) is an isometry, that is, for every \(x, y \in X, d(x, y)=d(f(x), f(y))\text{.}\) Prove that \(f(X)=X\text{.}\)
Solution.
Coming (not so) soon!
Suppose \(f:[0, \infty) \rightarrow \mathbb{R}\) is a uniformly continuous function. Prove there are real numbers \(A\) and \(B\) such that for every \(x \in[0, \infty)\text{,}\)
\begin{equation*} |f(x)| \leq A+B x. \end{equation*}
Solution.
Coming (not so) soon!
The following questions are not related.
  1. Let \(\alpha(x)=\left\{\begin{array}{ll}x & x<0 \\ e^{x} & x \geq 0\end{array}\right.\) Evaluate \(\int_{-1}^{1} x d \alpha\)
  2. Suppose \(f:[0,1] \rightarrow \mathbb{R}\) is continuous. Evaluate \(\lim _{n \rightarrow \infty} n \int_{0}^{1} x^{n} f(x) d x\text{.}\) Give a complete proof of your answer.
Solution.
Coming (not so) soon!
Suppose \(f:[0,1] \rightarrow[0,1]\) is a function such that \(f(0)=0, f(1)=1\text{,}\) has a continuous derivative, and \(f^{\prime}(x)>0\) for every \(x \in[0,1]\text{.}\) Prove there exists a sequence \(P_{n}\) of polynomials such that \(P_{n}\) converges uniformly to \(f\) and such that for every \(n\text{,}\) \(P_{n}\) is strictly increasing, \(P_{n}(0)=0\text{,}\) and \(P_{n}(1)=1\text{.}\)
Solution.
Coming (not so) soon!

Winter 2020.

Let \(\left\{a_{n}\right\}_{n=1}^{\infty} \subset(0, \infty)\) and \(c>0\) be given. Suppose that \(\lim _{n \rightarrow \infty} a_{n}=0\) and \(\sum_{n=1}^{\infty} a_{n}\) diverges. Prove that there exists a subsequence \(\left\{a_{n_{k}}\right\}_{k=1}^{\infty}\) such that \(\sum_{k=1}^{\infty} a_{n_{k}}=c\text{.}\)
Solution.
Coming (not so) soon!
Let \(\left\{a_{n}\right\}_{n=1}^{\infty},\left\{b_{n}\right\}_{n=1}^{\infty} \subset \mathbb{R}\) be bounded sequences, and define the sets
\begin{equation*} A:=\left\{a_{n}: n=1,2, \ldots\right\}, \quad B:=\left\{b_{n}: n=1,2, \ldots\right\}, \quad \text { and } \quad C:=\left\{a_{n}+b_{n}: n=1,2, \ldots\right\}. \end{equation*}
Prove or provide a counterexample each of the following statements.
  1. If \(a \in \mathbb{R}\) is a limit point for \(A\) and \(b \in \mathbb{R}\) is a limit point for \(B\text{,}\) then \(a+b\) is a limit point for \(C\text{.}\) (Here limit point means accumulation or cluster point.)
  2. If \(c \in \bar{C}\text{,}\) then there exists \(a \in \bar{A}\) and \(b \in \bar{B}\) such that \(a+b=c\text{.}\)
  3. If \(a_{n} \geq 0\) for all \(n=1,2, \ldots\text{,}\) then \(\limsup _{n \rightarrow \infty}\left(a_{n}^{2}\right)=\left(\limsup _{n \rightarrow \infty} a_{n}\right)^{2}\text{.}\)
Solution.
Coming (not so) soon!
Let \((X, \rho)\) be a metric space and define \(\sigma: X \times X \rightarrow[0, \infty)\) by
\begin{equation*} \sigma(x, y):=\min \{1, \rho(x, y)\}. \end{equation*}
  1. Prove that \(\sigma\) is a metric on \(X\text{.}\)
  2. Prove that \((X, \rho)\) is complete if and only if \((X, \sigma)\) is complete.
Solution.
Coming (not so) soon!
With \(a<b\text{,}\) let \(\mathcal{C}([a, b])\) denote the family of all \(\mathbb{R}\)-valued functions that are continuous on the interval \([a, b]\text{.}\)
  1. Let \(M<\infty\) and \(\mathcal{F} \subseteq \mathcal{C}([a, b])\) be given. Assume that each \(f \in \mathcal{F}\) is differentiable on \((a, b)\) and satisfies \(|f(a)| \leq M\) and \(\left|f^{\prime}(x)\right| \leq M\) for all \(x \in(a, b)\text{.}\) Prove that \(\mathcal{F}\) is equicontinuous on \([a, b]\text{.}\)
  2. Let a uniformly bounded sequence of functions \(\left\{g_{n}\right\}_{n=1}^{\infty} \subset \mathcal{C}([0,1])\) be given. For each \(n=1,2, \ldots\text{,}\) define \(f_{n}:[0,1] \rightarrow \mathbb{R}\) by
    \begin{equation*} f_{n}(x):= \begin{cases}0, & x=0 \\ \frac{1}{x} \int_{0}^{x} s g_{n}(s) \mathrm{d} s, & 0<x \leq 1.\end{cases} \end{equation*}
    Prove that there exists a subsequence of \(\left\{f_{n}\right\}_{n=1}^{\infty}\) that converges uniformly on \([0,1]\) to some \(f \in \mathcal{C}([0,1])\text{.}\)
Solution.
Coming (not so) soon!
  1. Let \(a, b \in \mathbb{R}\text{,}\) with \(a<b\text{,}\) be given, and suppose that \(f:(a, b) \rightarrow \mathbb{R}\) is differentiable on \((a, b)\) and that \(\lim _{x \rightarrow c} f^{\prime}(x)\) both exists and is finite, for all \(c \in(a, b)\text{.}\) Prove that \(f\) is continuously differentiable on \((a, b)\text{.}\)
  2. Produce a function \(f:(-1,1) \rightarrow \mathbb{R}\) that is everywhere differentiable and such that \(f^{\prime}\) is discontinuous at some \(c \in(-1,1)\text{.}\) Justify your claim.
Solution.
Coming (not so) soon!
The parts of this exploration are not connected.
  1. Let \(\left\{a_{n}\right\}_{n=1}^{\infty} \subset \mathbb{R}\) and a strictly increasing sequence \(\left\{x_{n}\right\}_{n=1}^{\infty} \subset(0,1)\) be given. Assume that \(\sum_{n=1}^{\infty} a_{n}\) is absolutely convergent, and define \(\alpha:[0,1] \rightarrow \mathbb{R}\) by
    \begin{equation*} \alpha(x):= \begin{cases}a_{n}, & x=x_{n} \\ 0, & \text { otherwise }.\end{cases} \end{equation*}
    Prove or disprove: \(\alpha\) has bounded variation on \([0,1]\text{.}\)
  2. Suppose that \(f:[0,1] \rightarrow \mathbb{R}\) is Riemann-Stieltjes integrable with respect to a nondecreasing function \(\beta:[0,1] \rightarrow[0, \infty)\text{.}\) Prove that \(f\) is Riemann-Stieltjes integrable with respect to the function \(\beta^{2}\text{.}\)
Solution.
Coming (not so) soon!

Spring 2019.

Let \(\left(M, d_{M}\right)\) and \(\left(N, d_{N}\right)\) be metric spaces. Define \(d_{M \times N}:(M \times N) \times(M \times N) \rightarrow \mathbb{R}\) by \(d_{M \times N}\left(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)\right):=d_{M}\left(x_{1}, x_{2}\right)+d_{N}\left(y_{1}, y_{2}\right), \quad\) for all \(x_{1}, x_{2} \in M\) and \(y_{1}, y_{2} \in N\text{.}\)
  1. Prove that \(\left(M \times N, d_{M \times N}\right)\) is a metric space.
  2. Let \(S \subseteq M\) and \(T \subseteq N\) be compact sets in \(\left(M, d_{M}\right)\) and \(\left(N, d_{N}\right)\text{,}\) respectively. Prove that \(S \times T\) is a compact set in \(\left(M \times N, d_{M \times N}\right)\text{.}\)
Solution.
Coming (not so) soon!
Let \(\left\{a_{n}\right\}_{n=1}^{\infty} \subset(0, \infty)\) be given, and assume that \(\sum_{n=1}^{\infty} a_{n}\) converges.
  1. Show that
    \begin{equation*} \sum_{n=1}^{\infty} \frac{a_{n}}{1+a_{n}} \text { converges and } \sum_{n=1}^{\infty} \frac{1}{1+a_{n}} \text { diverges. } \end{equation*}
  2. Suppose that \(\left\{b_{n}\right\}_{n=1}^{\infty} \subset \mathbb{R}\) satisfies \(\left|b_{n+1}-b_{n}\right| \leq a_{n}\text{,}\) for every \(n \in \mathbb{N}\text{.}\) Prove \(\left\{b_{n}\right\}_{n=1}^{\infty}\) is convergent.
Solution.
Coming (not so) soon!
Let \(f:[0, \infty) \rightarrow \mathbb{R}\) be continuous on \([0, \infty)\text{,}\) and define \(F:(0, \infty) \rightarrow \mathbb{R}\) by
\begin{equation*} F(x):=\sup _{0 \leq s<x} f(s)=\operatorname{lub}\{f(s): 0 \leq s<x\}. \end{equation*}
  1. Show that \(\lim _{x \rightarrow 0^{+}} F(x)\) exists.
  2. Prove that \(F\) is continuous on \((0, \infty)\text{.}\)
  3. Suppose that \(f\) is bounded, as well as continuous, on \([0, \infty)\text{.}\) Prove that \(F\) is uniformly continuous on \([0, \infty)\text{.}\)
Solution.
Coming (not so) soon!
Consider the formal series
\begin{equation*} \sum_{n=0}^{\infty} \mathrm{e}^{-2 n x}\left(1-\mathrm{e}^{-x}\right). \end{equation*}
Prove or disprove each of the following statements.
  1. The series converges uniformly on the interval \([1, \infty)\text{.}\)
  2. The series converges uniformly on the interval \([0,1]\text{.}\)
Solution.
Coming (not so) soon!
  1. Let \(\alpha:[0,1] \rightarrow \mathbb{R}\) be an increasing function that is continuous at 0, and let \(f:[0,1] \rightarrow \mathbb{R}\) be a bounded function. Suppose that, for each \(0<c<1\text{,}\) the function \(f\) is Riemann-Stieltjes integrable with respect to \(\alpha\) over \([c, 1]\text{.}\) Prove that \(f\) is Riemann-Stieljes integrable with respect to \(\alpha\) over \([0,1]\text{.}\)
  2. Let \(\left\{q_{n}\right\}_{n=1}^{\infty} \subset[0,1]\) be an enumeration of \(\mathbb{Q} \cap[0,1]\text{,}\) the rational numbers in \([0,1]\text{.}\) Define \(\beta:[0,1] \rightarrow \mathbb{R}\) by
    \begin{equation*} \beta(x):= \begin{cases}\frac{1}{n}, & x=q_{n}, \\ 0, & x \notin \mathbb{Q} \cap[0,1].\end{cases} \end{equation*}
    Prove or disprove: \(\beta\) has bounded variation on \([0,1]\text{.}\)
Solution.
Coming (not so) soon!
Suppose that \(f:[0,2] \rightarrow \mathbb{R}\) has the following properties:
  • \(f\) is continuous on \([0,2]\) and differentiable on \((0,2)\text{;}\)
  • \(\displaystyle f(0)=f(2)=0\)
  • there exists \(x_{0} \in(0,2)\) such that \(\left|f\left(x_{0}\right)\right| \geq 1\text{.}\)
  1. Prove that there exists \(c \in(0,2)\) such that \(\left|f^{\prime}(c)\right| \geq 1\text{.}\)
  2. There exists \(k>1\) such that for each \(\gamma \in[0, k]\text{,}\) there exists some \(c_{\gamma} \in(0,2)\) such that \(\left|f^{\prime}\left(c_{\gamma}\right)\right|=\gamma\text{.}\)
Solution.
Coming (not so) soon!

Winter 2019.

  1. Let \(f_{n}(x)=\frac{1}{1+n^{2} x^{2}}\) and \(g_{n}(x)=n x(1-x)^{n}\) for \(x \in[0,1]\text{.}\) Prove that \(\left\{f_{n}\right\}\) and \(\left\{g_{n}\right\}\) converge pointwise but not uniformly on \([0,1]\text{.}\)
  2. Are the families \(\left\{f_{n}\right\}\text{,}\) respectively \(\left\{g_{n}\right\}\) given in part (a) equicontinuous? Clearly motivate your answer.
Solution.
Coming (not so) soon!
Consider the following subset of the metric space \(\left(\mathcal{C}_{\mathrm{b}}([0,1]), \rho_{\infty}\right)\text{:}\)
\begin{equation*} A:=\left\{f \in \mathcal{C}_{\mathrm{b}}([0,1]): f([0,1]) \subseteq[0,1]\right\}. \end{equation*}
  1. Determine whether \(A\) is bounded, and if so what is its diameter.
  2. Determine whether \(A\) is closed in \(\mathcal{C}_{\mathrm{b}}([0,1])\text{.}\)
  3. Determine whether \(A\) is compact in \(\mathcal{C}_{\mathrm{b}}([0,1])\text{.}\)
Solution.
Coming (not so) soon!
Determine the values of \(x \in \mathbb{R}\) for which
\begin{equation*} \sum_{n=1}^{\infty} \frac{x^{n}}{1+n|x|^{n}} \end{equation*}
converges.
Solution.
Coming (not so) soon!
Suppose that \(f:[0,1] \rightarrow \mathbb{R}\) is differentiable and \(f(0)=0\text{.}\) Assume that there is a \(k>0\) such that
\begin{equation*} \left|f^{\prime}(x)\right| \leq k|f(x)| \end{equation*}
for all \(x \in[0,1]\text{.}\) Prove that \(f(x)=0\) for all \(x \in[0,1]\text{.}\)
Solution.
Coming (not so) soon!
Use the Riemann condition to show that \(f \in \mathcal{R}_{\alpha}[0,4]\) where \(f(x)=e^{2 x}\) and
\begin{equation*} \alpha(x)=\left\{\begin{array}{lc} x+1, & 0 \leq x \leq 2 \\ 3 x-2, & 2<x \leq 4 \end{array}\right. \end{equation*}
Compute the value of the Riemann-Stieltjes integral \(\int_{0}^{4} f(x) d \alpha\text{.}\)
Solution.
Coming (not so) soon!
Use the Heine-Borel Theorem to prove that if \(f\) is continuous on \([a, b]\) and \(f(x)>0\) for every \(x \in[a, b]\) then there exists \(\varepsilon>0\) such that \(f(x) \geq \varepsilon\) for every \(x \in[a, b]\text{.}\)
Solution.
Coming (not so) soon!

Spring 2018.

Determine whether the following sequences converge and carefully justify your claims:
  1. \(\displaystyle x_{n}=\frac{2 n \cdot n !}{n^{n}} ;\)
  2. \(y_{n}=\sum_{k=1}^{n} \frac{\cos (k !)}{k(k+1)}\text{.}\)
Solution.
Coming (not so) soon!
Find the domain of convergence and the sum of the series
\begin{equation*} \sum_{n \geq 0}(-1)^{n} \frac{x^{2 n+1}}{2 n+1} \end{equation*}
Show how one may use the sum of the series to provide an approximation for \(\pi\) up to three decimals. Be sure to provide all technical details.
Solution.
Coming (not so) soon!
Define \(f, \alpha \in \mathcal{B}([-2,2])\) by
\begin{equation*} f(x):=\left\{\begin{array}{ll} -1, & x \in[-2,0) ; \\ 3, & x \in[0,2] \end{array} \quad \text { and } \quad \alpha(x):= \begin{cases}-2, & x \in[-2,0] \\ 1, & x \in(0,2]\end{cases}\right. \end{equation*}
Determine whether \(f\) is Riemann-Stieltjes integrable with respect to \(\alpha\) over \([-1,1]\text{.}\) If it is, evaluate \(\int_{-1}^{1} f(x) \mathrm{d} \alpha(x)\)
Solution.
Coming (not so) soon!
  1. Given a set \(S\text{,}\) show that the function \(\rho_{\infty}: \mathcal{B}(S) \times \mathcal{B}(S) \rightarrow \mathbb{R}\) defined by
    \begin{equation*} \rho_{\infty}(f, g):=\operatorname{lub}(\{|f(x)-g(x)|: x \in S\}) \end{equation*}
    is a metric on \(\mathcal{B}(S)\text{.}\)
  2. Let \(M>0\) be given. Set
    \begin{equation*} S:=\left\{f \in \mathcal{C}_{\mathrm{b}}([0,1]): f(0)=0, f \text { is differentiable on }(0,1) \text {, and }\left|f^{\prime}(x)\right| \leq M \text { for each } x \in(0,1)\right\} \end{equation*}
    Determine whether the set \(S\) is compact in \(\left(\mathcal{C}_{\mathrm{b}}([0,1]), \rho_{\infty}\right)\text{.}\)
Solution.
Coming (not so) soon!
  1. If \(\left\{f_{n}\right\}_{n \geq 1}\) is an equicontinuous sequence of functions on a compact interval and \(f_{n} \rightarrow f\) pointwise, prove that the convergence is uniform.
  2. Let \(\alpha, M>0\) be given, and suppose that \(\left\{f_{n}\right\}_{n \geq 1}\) satisfies \(\left|f_{n}(x)-f_{n}(y)\right| \leq M|x-y|^{\alpha}\text{,}\) for all \(n \geq 1\) and all \(x, y\) in an interval \([a, b]\text{.}\) Show that \(\left\{f_{n}\right\}_{n \geq 1}\) is equicontinuous.
Solution.
  1. Claim: \((f_n)_{n=1}^\infty\) is bounded. If not, there is a subsequence \((f_{n_k})_{k=1}^\infty\) of \((f_n)_{n=1}^\infty\) so that \(\lim_{k\rightarrow\infty}\left|{\left|{{f_{n_k}}}\right|}\right|=\infty\text{.}\) By the extreme value theroem, exists \(x_k\in[a,b]\) with \(\left|{f_{n_k}(x_k)}\right|=\left|{\left|{{f_{n_k}}}\right|}\right|_\infty\text{.}\) Using that \([a,b]\) is compact and passing to a subsequence, we may assume \((x_k)_{k=1}^\infty\) converges to some \(x\in[a,b]\text{.}\) We'll show that \(\left|{f_{n_k}(x_k)}\right|\rightarrow\left|{f(x)}\right|\text{.}\) To this end, fix \(\varepsilon>0\text{.}\) Let \(\delta>0\) be such that for all \(k\geq 1, x,y\in[a,b]\) with \(\left|{x-y}\right|<\delta,\) we have
    \begin{equation*} \left|{f_{n_k}(x)-f_{n_k}(y)}\right|<\varepsilon. \end{equation*}
    Then choose \(L\geq 1\) so that for all \(k\geq L\text{,}\) we have \(d(x_k,x)<\delta\) and \(\left|{f_{n_k}(X)-f(x)}\right|<\varepsilon\text{.}\) Then if \(k\geq L\text{,}\) we have
    \begin{equation*} \begin{aligned} \left|{f(x_k)-f(x)}\right| <\left|{f_{n_k}(x_k)-f_{n_k}(x)}\right|+\left|{f_{n_k}(x)-f(x)}\right| <2\varepsilon \end{aligned} \end{equation*}
    where \(\left|{f_{n_k}(x_k)-f_{n_k}(x)}\right|<\varepsilon\) since \(d(x_k,x)<\delta\) (as \(k\geq L\)) and \(\left|{f_{n_k}(x)-f(x)}\right|<\varepsilon\) since \(k\geq L\text{.}\) Then
    \begin{equation*} \begin{aligned} f(x) =\lim_{k\rightarrow\infty}f_{n_k}(x_k). \end{aligned} \end{equation*}
    In particular,
    \begin{equation*} \begin{aligned} \left|{f(x)}\right| =\lim_{k\rightarrow\infty}\left|{f_{n_k}(x_k)}\right| =\lim_{k\rightarrow\infty}\left|{\left|{{f_{n_k}}}\right|}\right|_\infty =\infty, \end{aligned} \end{equation*}
    a contradiction.
    Suppose \((f_n)_{n=1}^\infty\) does not converge uniformly to \(f\text{.}\) Then exists \(\varepsilon>0\) such that for all \(N\geq 1,\exists n\geq N\) such that \(\left|{\left|{{f-f_n}}\right|}\right|_\infty\geq \varepsilon\text{.}\) Let \(\varepsilon>0\) as above. Then exists subsequence \((f_{n_k})_{k=1}^\infty\) so that for all \(k\geq 1\text{,}\) \(\left|{\left|{{f-f_{n_k}}}\right|}\right|_\infty\geq \varepsilon\text{.}\) Then \((f_{n_k})_{k=1}^\infty\) is bounded and equicontinuous. By the Arzelá-Ascoli Theorem, \((f_{n_k})_{k=1}^\infty\) has a subsequence \((f_{n_{k_l}})_{l=1}^\infty\) converging to some \(g\in C[a,b]\text{.}\) Then by above, we have \(\left|{\left|{{f-g}}\right|}\right|_\infty\geq \varepsilon\text{.}\) But \((f_{n_{k_l}})_{l=1}^\infty\) converge pointwise to both \(f\) and \(g\) so \(f=g\) and \(\left|{\left|{{f-g}}\right|}\right|_\infty=0\text{,}\) a contradiction.
Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a continuous function, and for all \(n \geq 1\text{,}\) put \(f_{n}(x)=f\left(x+\frac{1}{n}\right)\text{.}\)
  1. Show that \(f_{n}\) converges uniformly, as \(n \rightarrow \infty\text{,}\) over any closed interval \([a, b]\text{.}\)
  2. Give an example of a continuous function \(f\) for which the convergence is not uniform on \(\mathbb{R}\text{.}\)
Solution.
Coming (not so) soon!

Winter 2018.

Let \(\alpha\) be a monotonically increasing function on an interval \([a, b]\) and assume \(\alpha^{\prime}\) is Riemann integrable in \([a, b]\text{.}\) Let \(f\) be a bounded real function on \([a, b]\text{.}\) Prove that \(f \in \mathcal{R}(\alpha)\) implies \(f \alpha^{\prime} \in \mathcal{R}[a, b]\text{,}\) and
\begin{equation*} \int_{a}^{b} f d \alpha=\int_{a}^{b} f \alpha^{\prime} d x \end{equation*}
Solution.
Coming (not so) soon!
Let \(X\) be a metric space in which every infinite subset has a limit point. Prove that there is a countable subset \(D\) of \(X\) that is dense in \(X\text{.}\)
Solution.
Coming (not so) soon!
Let \(\left\{a_{n}^{\prime}\right\}\) be any rearrangement of an infinite sequence \(\left\{a_{n}\right\}\text{.}\) Assume \(\sum_{n=1}^{\infty} a_{n}\) converges absolutely. Prove \(\sum_{n=1}^{\infty} a_{n}^{\prime}=\sum_{n=1}^{\infty} a_{n}\text{.}\)
Solution.
Coming (not so) soon!
Suppose \(f_{n} \rightarrow f\) uniformly on \([0, \infty)\) and \(\lim _{x \rightarrow \infty} f_{n}(x)\)
\begin{equation*} \lim _{x \rightarrow \infty} \lim _{n \rightarrow \infty} f_{n}(x)=\lim _{n \rightarrow \infty} \lim _{x \rightarrow \infty} f_{n}(x) \end{equation*}
Solution.
Coming (not so) soon!
Let \(f:[a, b] \rightarrow \mathbb{R}\text{.}\) Suppose \(f \in B V[a, b]\text{.}\) Prove \(f\) is the difference of two increasing functions.
Solution.
Coming (not so) soon!
Suppose that \(f:[0,2] \rightarrow \mathbb{R}\) is continuous on \([0,2]\text{,}\) differentiable on \((0,2)\text{,}\) satisfying \(f(0)=\) \(f(2)=0\) and \(f(c)=1\) for some \(c \in(0,2)\text{.}\) Prove that there is an \(x \in(0,2)\) such that \(\left|f^{\prime}(x)\right|>1\text{.}\)
Solution.
Coming (not so) soon!

Spring 2017.

Suppose that \(f: \mathbb{R} \rightarrow \mathbb{R}\) is given by: \(f(x)=\frac{x^{3}}{1+x^{2}}\text{.}\) By using the \(\epsilon-\delta\) definition only, prove that \(f\) is uniformly continuous on \(\mathbb{R}\text{.}\)
Hint.
Hint: write \(f(x)=x-\frac{x}{1+x^{2}}\text{.}\)
Solution.
Coming (not so) soon!
Let \((X, d)\) be a metric space. A function \(f: X \rightarrow \mathbb{R}\) is said to be lower semicontinuous at \(x_{0} \in X\) if, for every \(\epsilon>0\) there exists \(\delta>0\) such that for all \(x \in X\) with \(d\left(x, x_{0}\right)<\delta\text{,}\) then \(f(x)-f\left(x_{0}\right)>-\epsilon\text{.}\) Prove: \(f\) is lower semicontinuous at \(x_{0} \in X\) if and only if \(\liminf _{n \rightarrow \infty} f\left(x_{n}\right) \geq f\left(x_{0}\right)\) whenever \(\left\{x_{n}\right\}_{n=1}^{\infty} \subset X\) satisfies \(\lim _{n \rightarrow \infty} x_{n}=x_{0}\) in \(X\text{.}\)
Solution.
Coming (not so) soon!
Let \((X, d)\) be a compact metric space. Suppose that \(f_{n}: X \rightarrow[0, \infty)\) is a sequence of continuous functions with \(f_{n}(x) \geq f_{n+1}(x)\text{,}\) for all \(n \in \mathbb{N}\) and all \(x \in X\text{,}\) and such that \(f_{n} \longrightarrow 0\) point-wise on \(X\text{.}\) Prove \(\left\{f_{n}\right\}_{n=1}^{\infty}\) converges to zero uniformly on \(X\text{.}\)
Solution.
Coming (not so) soon!
Let \(f:[-1,1] \longrightarrow \mathbb{R}\) be a function such that \(f \in C^{2}([-1,1])\text{,}\) i.e., \(f\) is twice-continuously differentiable on \([-1,1]\text{,}\) and \(f^{\prime \prime \prime}\) exists on \((-1,1)\) with \(\left|f^{\prime \prime \prime}(x)\right| \leq 1\) for all \(x \in(-1,1)\text{.}\) Let:
\begin{equation*} a_{n}=n\left(f\left(\frac{1}{n}\right)-f\left(\frac{-1}{n}\right)\right)-2 f^{\prime}(0). \end{equation*}
Prove that the series \(\sum_{n=1}^{\infty} a_{n}\) converges absolutely.
Solution.
Coming (not so) soon!
Suppose that \(f:[0,1] \rightarrow \mathbb{R}\) is a continuous function which satisfies, for \(n=0,1,2, \ldots\text{,}\)
\begin{equation*} \int_{0}^{1} f(\sqrt[2 n+1]{x}) d x=0 \end{equation*}
Prove that \(f(x)=0\) for all \(x \in[0,1]\text{.}\)
Solution.
Coming (not so) soon!
For each \(n \in \mathbb{N}\text{,}\) let \(f_{n}:[0, \infty) \longrightarrow \mathbb{R}\) be given by: \(f_{n}(x)=\frac{\sin (n x)}{1+n x}\text{.}\)
  1. Prove \(f_{n}\) converges point-wise on \([0, \infty)\) and find the point-wise limit \(f\text{.}\)
  2. Show that \(f_{n} \longrightarrow f\) uniformly on \([a, \infty)\) for every \(a>0\text{,}\) but the convergence is not uniform on \([0, \infty)\text{.}\)
Solution.
Coming (not so) soon!

Winter 2017.

Consider the series \(\sum_{n=1}^{\infty} \frac{(-1)^{n}}{1+n x}, x \in \mathbb{R}\text{.}\) Justify your answers to the following questions:
  1. Does the series converge at \(x=0\text{?}\)
  2. Does the series converge on \((0, \infty)\text{?}\)
  3. Does the series converge uniformly on \((0, \infty)\text{?}\)
  4. Does the series converge uniformly on \([a, \infty)\) for any \(a>0\text{.}\)
Solution.
Coming (not so) soon!
Prove: \(f \in \mathcal{R}(\alpha)\) on \([a, b] \Longleftrightarrow\) For any \(a<c<b, f \in \mathcal{R}(\alpha)\) on \([a, c]\) and on \([c, b]\text{.}\) In addition, if either condition holds, then we always have
\begin{equation*} \int_{a}^{c} f d \alpha+\int_{c}^{b} f d \alpha=\int_{a}^{b} f d \alpha \end{equation*}
Solution.
Coming (not so) soon!
Let \(f(x)=\sum_{n=0}^{\infty} a_{n}(1-x) x^{n}\text{,}\) where \(\left\{a_{n}\right\}_{n=0}^{\infty} \subset \mathbb{R}\text{.}\) Assume \(\lim _{n \rightarrow \infty} a_{n}=a \in \mathbb{R}\text{.}\)
  1. Prove: \(f(x)\) converges absolutely for \(|x|<1\text{.}\)
  2. Prove: \(\lim _{x \rightarrow 1^{-}} f(x)=a\text{.}\)
Solution.
Coming (not so) soon!
Let \(f \in C[0,1]\text{.}\) Prove that there is a sequence of polynomials, \(\left\{p_{n}\right\}\text{,}\) of even degree monomials (i.e. every term in \(p_{n}\) is of the form \(a_{k} x^{2 k}\) for integer \(k \geq 0\) ) that converges to \(f\) uniformly on \([0,1]\text{.}\)
Solution.
Coming (not so) soon!
Assume \(\left\{f_{n}\right\}\) is a sequence of monotonically increasing functions on \([a, b]\text{,}\) and \(f_{n} \rightarrow f\) point-wise. Prove that if \(f\) is continuous then the convergence \(f_{n} \rightarrow f\) is uniform on \([a, b]\text{.}\)
Solution.
Coming (not so) soon!
Let \((X, d)\) be a metric space. A function \(f: X \rightarrow \mathbb{R}\) is said to be lower semi-continuous (l.s.c.) if \(f^{-1}(a, \infty)=\{x \in X: f(x)>a\}\) is open in \(X\) for every \(a \in \mathbb{R}\text{.}\) Analogously, \(f\) is upper semi-continuous (u.s.c.) if \(f^{-1}(-\infty, b)=\{x \in X: f(x)<b\}\) is open in \(X\) for every \(b \in \mathbb{R}\text{.}\)
  1. Prove that a function \(f: X \rightarrow \mathbb{R}\) is continuous \(\Longleftrightarrow f\) is both l.s.c and u.s.c.
  2. Prove that \(f\) is lower semi-continuous (l.s.c.) \(\Longleftrightarrow \liminf _{n \rightarrow \infty} f\left(x_{n}\right) \geq f(x)\) whenever \(\left\{x_{n}\right\}_{n=1}^{\infty} \subset X\) such that \(x_{n} \rightarrow x\) in \(X\text{.}\)
Solution.
Coming (not so) soon!

Spring 2016.

Consider the function: \(f(x)=\frac{x}{1-x^{2}}, \quad x \in(0,1)\text{.}\)
  1. By using the \(\epsilon-\delta\) definition of the limit only, prove that \(f\) is continuous on \((0,1)\text{.}\) (Note: You are not allowed to trivialize the problem by using properties of limits).
  2. Is \(f\) uniformly continuous on \((0,1)\text{?}\) Justify your answer.
Solution.
Coming (not so) soon!
Let \(\left\{a_{k}\right\}_{k=1}^{\infty}\) be a bounded sequence of real numbers and \(E\) be given by:
\begin{equation*} E:=\left\{s \in \mathbb{R}: \text { the set }\left\{k \in \mathbb{N}: a_{k} \geq s\right\} \text { has at most finitely many elements }\right\}. \end{equation*}
Prove that \(\limsup _{k \rightarrow \infty} a_{k}=\inf E\text{.}\)
Solution.
Coming (not so) soon!
Assume \((X, d)\) is a compact metric space.
  1. Prove that \(X\) is both complete and separable.
  2. Suppose \(\left\{x_{k}\right\}_{k=1}^{\infty} \subset X\) is a sequence such that the series \(\sum_{k=1}^{\infty} d\left(x_{k}, x_{k+1}\right)\) is convergent. Prove that the sequence \(\left\{x_{k}\right\}_{k=1}^{\infty}\) converges in \(X\text{.}\)
Solution.
Coming (not so) soon!
Suppose that \(f:[0,2] \rightarrow \mathbb{R}\) is continuous on \([0,2]\text{,}\) differentiable on \((0,2)\text{,}\) and such that \(f(0)=f(2)=0, f(c)=1\text{,}\) for some \(c \in(0,2)\text{.}\) Prove that there is an \(x \in(0,2)\) such that \(\left|f^{\prime}(x)\right|>1\)
Solution.
Coming (not so) soon!
Let \(f_{n}(x)=n^{\beta} x\left(1-x^{2}\right)^{n}, x \in[0,1], n \in \mathbb{N}\text{.}\)
  1. Prove that \(\left\{f_{n}\right\}_{n=1}^{\infty}\) converges point-wise on \([0,1]\) for every \(\beta \in \mathbb{R}\text{.}\)
  2. Show that the convergence in part (a) is uniform on \([0,1]\) for all \(\beta<\frac{1}{2}\text{,}\) but not uniform for any \(\beta \geq \frac{1}{2}\text{.}\)
Solution.
Coming (not so) soon!
  1. Suppose \(f:[-1,1] \longrightarrow \mathbb{R}\) is a bounded function that is continuous at 0. Let \(\alpha(x)=-1\) for \(x \in[-1,0]\) and \(\alpha(x)=1\) for \(x \in(0,1]\text{.}\) Prove that \(f \in \mathcal{R}(\alpha)[-1,1]\text{,}\) i.e., \(f\) is Riemann integrable with respect to \(\alpha\) on \([-1,1]\text{,}\) and \(\int_{-1}^{1} f d \alpha=2 f(0)\text{.}\)
  2. Let \(g:[0,1] \longrightarrow \mathbb{R}\) be a continuous function such that \(\int_{0}^{1} g(x) x^{3 k+2} d x=0\text{,}\) for all \(k=0,1,2, \cdots\text{.}\) Prove that \(g(x)=0\) for all \(x \in[0,1]\text{.}\)
Solution.
Coming (not so) soon!
Recall this is the set obtained by removing \((1 / 3,2 / 3)\) from \([0,1]\text{,}\) then removing the middle third from the remaining intervals, etc.
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