$$ \newcommand{\norm}[1]{\left\|#1\right\|} \newcommand{\paren}[1]{\left(#1\right)} \newcommand{\abs}[1]{\left\lvert#1\right\rvert} \newcommand{\set}[1]{\left\{#1\right\}} \newcommand{\ang}[1]{\left\langle#1\right\rangle} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\N}{\mathbb{N}} $$

Math 142B: Introduction to Analysis II

Lecture B (Charlesworth)

Last modified: February 21, 2018.


Course Information


Homework assignments will be available on this webpage throughout the term. All homework assignments must be submitted to the drop boxes in the basement of AP&M by 4:00 PM on the deadline.

    1. Show that if $\mathcal{P_1}, \ldots, \mathcal{P_n}$ are partitions of the interval $[a, b]$, then there is a partition $\mathcal{P}$ of the interval $[a,b]$ which is a refinitement of every $\mathcal{P}_i$; that is, show any finite set of partitions has a common refinement.
    2. Suppose $\paren{\mathcal{P_n}}_n$ is an infinite sequence of partitions of $[a, b]$. Must there be a partition which is a refinement of every partition in the sequence? (Make sure to check the definition of "partition" carefully.)
  1. Let $f : [0, 2] \to \R$ be defined by $f(t) = t$. Find a partition $\mathcal{P}$ of $[0, 2]$ such that $$U(f, \mathcal{P}) - L(f, \mathcal{P}) \lt \frac{1}{256}.$$
    1. Suppose that $f : [a,b] \to \R$ is integrable, and $g : [a,b]\to\R$ is such that $g(x) = f(x)$ for all $x \in [a,b] \setminus\set{x_0}$. Show that $g$ is integrable, and $$\int_a^b f(t)\,dt = \int_a^b g(t)\,dt.$$
    2. Using the previous part, show that if $f : [a,b] \to \R$ is integrable and $g : [a,b]\to\R$ is such that $g(x) = f(x)$ for all but finitely many points in $[a, b]$ then $g$ is integrable, and $$\int_a^b f(t)\,dt = \int_a^b g(t)\,dt.$$ (Hint: use induction.)
    3. Suppose that $f : [a, b] \to \R$ is integrable, and $g : [a,b] \to \R$ is such that $g(x) = f(x)$ except for at a sequence of points $(x_n)_n$. Must $g$ be integrable? Prove it, or provide a counterexample.
  2. Suppose $f : [a,b] \to \R$ is bounded, $c \in (a, b)$, and $f$ is integrable on both $[a, c]$ and $[c, b]$. Show that $f$ is integrable on $[a, b]$.
    1. Find bounded functions $f, g : [0, 1] \to \R$ so that $f + g$ is integrable, but $f$ is not.
    2. Find bounded functions $f, g : [0, 1] \to (0, \infty)$ so that $fg$ is integrable, but $f$ is not.
  3. Bonus:

    Let the function $f$ be defined as follows: $$\begin{align*} f : [0,1] &\longrightarrow \R\\ x &\longmapsto \begin{cases}\frac1q & \text{ if } x = \frac{p}{q} \in \Q \text{ with } p, q \in \N, \gcd(p,q) = 1{\color{red},} \text{ and } x \neq 0\\ 0 & \text{ otherwise.} \end{cases} \end{align*}$$ Show that $f$ is integrable, and find its integral.

    Notice that $\frac{d}{dt}\int_0^t f(x)\,dx \neq f(t)$.

Assignment 1, due January 19th, 2018.
    1. Suppose that $f : [a,b]\to\R$ is integrable. Show that $|f|$ is integrable. (Hint: notice that for any $x, y \in [a,b]$, $|f(x)| - |f(y)| \leq |f(x)-f(y)|$.)
    2. Suppose that $f, g : [a,b]\to\R$ are integrable. Show that $\min(f, g)$ and $\max(f,g)$ are integrable. (Hint: there is a reason that this is Part B.)
  1. Suppose $f : [a,b]\to[0,\infty)$ is continuous. Show that $\int_a^b f = 0$ if and only if $f(t) = 0$ for all $t \in [a,b]$.
  2. Suppose $f$ is defined as follows: $$\begin{align}f : [4, 12] &\longrightarrow \R\\ t &\longmapsto \begin{cases} 3 & \text{ if } 5 \lt t \leq 10 \\ 1 & \text{ otherwise.}\end{cases}\end{align}$$ Show that $f$ does not have an antiderivative.

    (Hint: There are several ways of solving this problem. One may take a direct approach: first, show that if $f$ does have an antiderivative, then it must be a piecewise linear function with different slopes on the different pieces; but there is no way such a function could be differentiable where the slope changes. In fact, it is possible to prove a much more general statement: any function with a jump discontinuity has no antiderivative.)

  3. Suppose that $f : \R \to \R$ has a continuous second derivative.
    1. Show that $f(t) = f(0) + f'(0)t + \int_0^t (t-x)f''(x)\,dx$ for all $t$. (Hint: compare the second derivatives of both sides.)
    2. Show that if $G : \R \to \R$ is defined by $G(t) = \int_0^t(t-x)f(x)\,dx$, then $G''(t) = f(t)$ for all $t \in \R$.
  4. Suppose that $f : [a, b]\to\R$ is continuous, and that $F : [a,b] \to \R$ is continuous, differentiable on $(a,b)$, and $F'(z) = f(z)$ for $z \in (a,b)$. Use the second fundamental theorem to prove the first fundamental theorem by showing that $$\frac{d}{dt}\paren{F(t) - \int_a^t f(x)\,dx} = 0.$$


    First, show the identity above. Then, use this to prove the first fundamental theorem: if $F : [a, b] \to \R$ is continuous, differentiable on $(a, b)$, and $F'$ is continuous and bounded on $(a, b)$ then $$\int_a^b F'(x)\,dx = F(b) - F(a).$$

  5. Bonus:

    A set $S \subset \R$ is said to be null if for any $\epsilon \gt 0$ there is a sequence of open intervales $\paren{(a_n, b_n)}_n$ such that their union contains $S$ and their combined length is less than $\epsilon$: $$S\subseteq \bigcup_{n=1}^\infty (a_n, b_n) \qquad\qquad\text{ and }\qquad\qquad \sum_{n=1}^\infty b_n-a_n \lt \epsilon.$$

    1. Suppose that $f : [a, b] \to \R$ is bounded, and that $\set{x \in [a,b] : f \text{ is discontinuous at } x}$ is null. Show that $f$ is integrable.
    2. Show that if $S \subset \R$ is countable, then $S$ is null.

    As it turns out, there are null sets which are not countable; the Cantor set is a famous example. Somewhat more surprisingly, the function which is $1$ on the Cantor set and $0$ on its complement is discontinuous precisely on the Cantor set, and therefore integrable by this problem. On the last assignment's bonus problem we showed that there was an integrable function with a countable set of discontinuities; now there is even one with an uncountable set of discontinuities.

Assignment 2, due January 29th, 2018.
  1. Let's verify that $e^x$ can be defined with the usual properties that we expect. In this problem, be sure not to use any properties of the logarithm before establishing them. Define the function $L$ as follows: $$\begin{align*}L : (0, \infty) &\longrightarrow \R\\x&\longmapsto\int_1^x \frac1t\,dt.\end{align*}$$
    1. Explain why $L$ is differentiable.
    2. Show that $\displaystyle\lim_{t\to\infty}L(t) = \infty$. (Hint: compare $L(n)$ to the harmonic series $\sum \frac1k$, which we know diverges from 142A.)
    3. Show that $L(ab) = L(a) + L(b)$. (A question from the midterm may be useful here.)
    4. Conclude that $\displaystyle\lim_{t\to0^+}L(t) = -\infty$.
    5. The fact that $L$ is continuous means we know know that its range is $(-\infty, \infty)$. Show that $L$ is invertible. (Hint: there is a very useful criterion from 142A about monotonic functions and invertiblilty; it may be useful to show that $L$ is monotonic.)
    6. Define the function $E : \R \to (0, \infty)$ to be the inverse of $L$, so $E(L(x)) = x$ and $L(E(y)) = y$. Show that $E'(x) = E(x)$.
    7. Show that for any rational number $q \in \Q$, $E(q) = E(1)^q$. (Hint: use Part C; recall that in 142A we were able to prove that for any $\alpha \gt 0$ and any $x, y \in \R$, $\alpha^{x+y} = \alpha^x\alpha^y$.)
    8. Use the fact that for any $\alpha \gt 0$ the function $x \mapsto \alpha^x$ is continuous to explain why $E(x) = E(1)^x$ for all $x \in \R$.
    9. Show that the Taylor series for $E$ centred at $0$ converges to $E$ at every point.
    10. Conclude that $E(1) = \sum_{j=0}^\infty \frac1{j!}$. This number, $E(1)$, is the familiar constant $e$.
  2. Show that if $p$ is a polynomial such that $p(x_0) = 0$, then there is a polynomial $q$ so that $p(x) = (x-x_0)q(x)$. Conclude that a polynomial of degree $n$ has at most $n$ roots.
  3. Suppose that $f, g : I \to \R$ have $n+1$ continuous derivatives. Show that they have contact of order $n$ at $x_0 \in I$ if and only if $$\lim_{x\to x_0}\frac{f(x)-g(x)}{(x\color{red}{-x_0})^n} = 0.$$
  4. Bonus:

    Suppose that $R \gt 0$ and $f : (-R, R) \to \R$ is infinitely differentiable. Suppose further that for any $0 \lt r \lt R$ there is a sequence $(M_{r,k})_k$ such that for all $x \in (-r, r)$, $$\abs{f^{(k)}(x)} \leq M_{r,k} \qquad\qquad\text{and}\qquad\qquad \lim_{k\to\infty} M_{r,k}\frac{r^k}{k!} = 0.$$ Show that for every $x \in (-\color{red}{R}, \color{red}{R})$, $$\int_0^x f(t)\,dt = \sum_{k=1}^\infty f^{(k-1)}(0)\frac{x^k}{k!}.$$

    (Hint: first show that our assumptions bounding the derivatives of $f$ mean that the Taylor polynomials converge uniformly to $f$ in the sense of this 142A assignment; next show that if a sequence of functions $(f_n)$ converges uniformly to $f$ then the sequence of integrals $\int_a^b f_n$ converges to $\int_a^b f$; then put these two statements together.)

Assignment 3, due February 9th, 2018.
    1. Use Taylor polynomials to show that for every $n \in \N$ there is a constant $C \gt 0$ so that for every $x \geq 0$, $e^x \gt Cx^n$.
    2. Show that for every polynomial $p$, $$\lim_{x\to\infty}\frac{p(x)}{e^x} = 0.$$
  1. Suppose that $I, J$ are open intervals, $g : J \to I$, and $f : I \to \R$, so that $f\circ g : J \to \R$.
    1. How many derivatives of $f$ and how many of $g$ must exist for $f\circ g$ to be $n$ times differentiable?
    2. Let $x_0 \in J$, and find an expression for the $n$-th Taylor polynomial of $f\circ g$ at $x_0$ in terms of Taylor polynomials for $f$ and $g$. Be sure to specify where these polynomials are centred!
  2. Recall that (soon) we (will) have shown that the function $$f : x \longmapsto \begin{cases}e^{-\frac1{x^2}} & \text{ if } x \neq 0\\0&\text{ else}\end{cases}$$ is infinitely differentiable and has the property that $f^{(n)}(0) = 0$ for all $n$.
    1. Show that if $I$ is an open interval, there is a function $f : \R \to \R$ which is infinitely differentiable so that $f(x) = 0$ when $x \notin I$, and $f(x) \gt 0$ when $x \in I$.
    2. Use the above to show that if $J$ is a closed interval and $I$ is an open interval with $J \subset I$, there is a function $g : \R \to \R$ which is infinitely differentiable so that $g(x) = 0$ when $x \notin I$ and $g(x) = 1$ when $x \in J$.
  3. We developed the following estimates when examining the Taylor series for logarithm: $$\begin{align*}\abs{\ln(1+y)-p_n(y)} &\leq \frac{1}{1+y}\frac{\abs{y}^{n+1}}{n+1}&\qquad\qquad&\text{if } -1 \lt y \leq 0,\\ \abs{\ln(1+x)-p_n(x)} &\leq \frac{x^{n+1}}{n+1} &&\text{if } 0 \leq x \leq 1.\end{align*}$$
    1. Suppose $0 \lt x \lt 1$. Notice that we can approximate $\ln(1+x)$ by $p_n(x)$, or by choosing $y$ so that $\frac1{1+x} = 1+y$, we can approximate $\ln(1+x) = -ln(1+y)$ by $-p_n(y)$. Is either approximation better? Which one?
    2. How large must $n$ be for the approximation of $\ln(1.01)$ by $p_n(0.01)$ to be accurate to at least 8 digits? Using the above method, how large must $n$ be for the approximation via $-p_n(y)$ to be accurate to at least 8 digits?
  4. Show that the Weiestrass Approximation Theorem fails on sets which are not sequentially compact:
    1. Suppose $S \subset \R$ is unbounded. Show that $f(x) := e^{x^2}$ cannot be uniformly approximated by polynomials on $S$. (Hint: Problem 1 may be helpful. Be sure not to make assumptions about $S$ which may not be true.)
    2. Suppose $S' \subset \R$ is not closed. Find a function $g : S' \to \R$ which cannot be uniformly approximated by polynomials. (Hint: if $S'$ is not closed, there is a limit point of $S'$ which is not contained in $S'$.)
  5. Bonus:

    The following two problems are thematically related -- both relate to when continuous things cannot be approximated by polynomials -- but you should not assume that either will be useful for proving the other. The second problem is a result usually proved using sophisticated techniques from complex analysis which are not within the scope of this course. However, the result is possible to establish (though not stratightforward) using techniques available to us.

    1. Show that the only functions from $\R$ to $\R$ which can be uniformly approximated by polynomials are the polynomials.
    2. A complex number is something of the form $\alpha + \beta i$ where $\alpha, \beta \in \R$; the set of complex numbers is denoted $\C$, and one identifies $\R$ with the set $\set{x+0i : x \in \R} \subset \C$. Addition is performed by components (so $(a + bi) + (c + di) = (a+c) + (b+d)i$), while multiplication is distributed and it is understood that $i^2 = -1$ (so $(a + bi)(c + di) = (ac - bd) + (bc+ad)i$). If $z = a+bi$, its complex conjugate, denoted $\bar{z}$, is $\bar{z} = a-bi$. The norm of $z = a+bi$ is the quantity $|z| = \sqrt{a^2+b^2}$. It follows that $z\bar{z} = |z|^2 + 0i \in \R$. In turn it follows that if $|z| = 1$ then $z\bar{z} = 1$.

      A complex polynomial is a function $p : \C \to \C$ of the form $p(z) = a_0 + a_1z + a_2z^2 + \ldots + a_nz^n$, where $a_0, \ldots, a_n \in \C$.

      Let $\mathbb{D}$ be the set $\set{z \in \C : |z| \leq 1}$. Show that the function $f : \mathbb{D} \to \C$ given by $f(z) = \bar{z}$ is not uniformly approximated by polynomials: there is some $\epsilon \gt 0$ so that for any polynomial $p$ there is some $z \in \mathbb{D}$ so that $|f(z) - \bar{z}| \gt \epsilon$.

Assignment 4, due February 21st, 2018.

You may wish to type your homework; for example, this makes it much easier for others to read, and makes it easier to edit and produce a coherent final argument. Most modern mathematics papers are typeset using a system called \(\mathrm{\LaTeX}\) (pronounced "lah-tech" or "lay-tech"; see the Wikipedia entry on Pronouncing and writing "LaTeX"). Although it has a steep learning curve, it is extremely useful for typesetting complicated mathematical expressions. There are many resources available online, such as this reference by Oetiker, Partl, Hyna, and Schlegl. I have also made available an assignment template here, which produces this output when compiled correctly.

Working in groups on assignments is encouraged in this course. However, you should write your answers individually.

Instructional Staff

Ian Charlesworth Instructor AP&M 5880C ilc@math.ucsd.edu
Woonam Lim Teaching Assistant AP&M 6414 w9lim@ucsd.edu
Yucheng Tu Teaching Assistant AP&M 5720 y7tu@ucsd.edu

My office hours are Mondays from 10:00am - 11:00am, Wednesdays from 10:30am - 11:30am, and Fridays from 12:30pm - 1:30pm.

We will be communicating with you and making announcements through an online question and answer platform called Piazza. We ask that when you have a question about the class that might be relevant to other students, you post your question on Piazza instead of emailing us. That way, everyone can benefit from the response. Posts about homework or exams on Piazza should be content based. While you are encouraged to crowdsource and discuss coursework through Piazza, please do not post complete solutions to homework problems there. Questions about grades should be brought to the instructors, in office hours. You can also post private messages to instructors on Piazza, which we prefer over email.

If emailing us is necessary, please do the following:

Class Meetings

Lecture B00 (Charlesworth) Mondays, Wednesdays, Fridays 2:00pm - 2:50pm PCYNH 122
Discussion B01 (Woonam Lim) Tuesdays6:00pm - 6:50pmAP&M 2402
Discussion B02 (Yucheng Tu) Tuesdays7:00pm - 7:50pmAP&M 2402
Discussion B03 (Yucheng Tu) Tuesdays8:00pm - 8:50pmAP&M 2402
Final Exam Monday, March 19 3:00pm - 6:00pm TBA



Course:  Math 142B

Title:  Math 142B: Introduction to Analysis II

Credit Hours:  4  (Students may not receive credit for both Math 140B and 142B.)

Prerequisites:  Math 142A or Math 140A, or consent of instructor.

Catalog Description:  Second course in an introductory two-quarter sequence on analysis. Topics include: differentiation, the Rieman integral, sequences and series of functions, uniform convergence, Taylor and Fourier series, special functions. See the UC San Diego Course Catalog.

Textbook: Advanced Calculus, 2nd edition, by Patrick M. Fitzpatrick.

Subject Material:  We will mainly cover chapters 6, 8, and 9 of the text.

Lecture:  Attending the lecture is a fundamental part of the course; you are responsible for material presented in the lecture whether or not it is discussed in the textbook.  You should expect questions on the exams that will test your understanding of concepts discussed in the lecture.

Reading:  Reading the sections of the textbook corresponding to the assigned homework exercises is considered part of the homework assignment; you are responsible for material in the assigned reading whether or not it is discussed in the lecture.

Calendar of Lecture Topics:   The following calendar is subject to revision during the term. The section references are only a guide; our pace may vary from it somewhat.

Week Monday Tuesday Wednesday Thursday Friday
1 Jan 8
Jan 9
Jan 10
Jan 11 Jan 12
2 Jan 15
Martin Luther King, Jr. Holiday
Jan 16
Jan 17
Jan 18 Jan 19
3 Jan 22
Jan 23
Jan 24
Jan 25 Jan 26
4 Jan 29
Jan 30
Jan 31
Mid-term exam
Feb 1 Feb 2
5 Feb 5
Feb 6
Feb 7
Feb 8 Feb 9
6 Feb 12
Feb 13
Feb 14
Feb 15 Feb 16
7 Feb 19
Presidents' Day Holiday
Feb 20
Feb 21
Feb 22 Feb 23
Mid-term exam
8 Feb 26
Feb 27
Feb 28
Mar 1 Mar 2
9 Mar 5
Mar 6
Mar 7
Mar 8 Mar 9
10 Mar 12
Mar 13
Mar 14
Mar 15 Mar 16
11 Mar 19 Mar 20 Mar 21 Mar 22 Mar 23

Homework:  Homework is a very important part of the course and in order to fully master the topics it is essential that you work carefully on every assignment and try your best to complete every problem. Homework will be assigned on the course webpage. Your homework can be submitted to the dropbox with your TA's name on it in the basement of the AP&M building. Homework is officially due at 4:00 PM on the due date.

Midterm Exams:  There will be two midterm exams given during the quarter. You may bring one 8.5 by 11 inch sheet of paper with handwritten notes (on both sides) with you to each midterm exam; no other notes (or books) will be allowed. No calculators, phones, or other electronic devices will be allowed during the midterm exams.   You may bring at most three four-leaf clovers, horseshoes, maneki-neko, or other such talismans for good luck. There will be no makeup exams.

Final Examination:  The final examination will be held at the date and time stated above.

Administrative Links:    Here are two links regarding UC San Diego policies on exams:

Regrade Policy:  

Administrative Deadline:  Your scores for all graded work will be posted to TritonEd.

Grading: Your course grade will be determined by your cumulative average at the end of the term and will be based on the following scale:

A+ A A- B+ B B- C+ C C-
97 93 90 87 83 80 77 73 70
Your cumulative average will be the best of the following two weighted averages:

In addition,  you must pass the final examination in order to pass the course.  Note: Since there are no makeup exams, if you miss a midterm exam for any reason, then your course grade will be computed with the second option. There are no exceptions; this grading scheme is intended to accommodate emergencies that require missing an exam.

Your single worst homework score will be ignored.

Academic Integrity:  UC San Diego's code of academic integrity outlines the expected academic honesty of all studentd and faculty, and details the consequences for academic dishonesty. The main issues are cheating and plagiarism, of course, for which we have a zero-tolerance policy. (Penalties for these offenses always include assignment of a failing grade in the course, and usually involve an administrative penalty, such as suspension or expulsion, as well.) However, academic integrity also includes things like giving credit where credit is due (listing your collaborators on homework assignments, noting books or papers containing information you used in solutions, etc.), and treating your peers respectfully in class. In addition, here are a few of our expectations for etiquette in and out of class.


Students requesting accommodations for this course due to a disability must provide a current Authorization for Accommodation (AFA) letter issued by the Office for Students with Disabilities (OSD) which is located in University Center 202 behind Center Hall. Students are required to present their AFA letters to Faculty (please make arrangements to contact me privately) and to the OSD Liaison in the department in advance (by the end of Week 2, if possible) so that accommodations may be arranged. For more information, see here.



Here are some additional resources for this course, and math courses in general.

Lecture Notes

Any remarks pertaining to particular lectures will be posted here throughout the term.

Graphs of Bernstein polynomials.

Below is a graph of the functions $f_k(x) = \binom{10}{k}x^k(1-x)^{10-k}$ for $0 \leq k \leq 10$.

Below is a graph of the functions $g_k(x) = \sin\paren{2\pi\frac{k}{10}}f_k(x)$, as well as their sum, and the function $\sin(2\pi x)$.

CSS and page template greatfully taken from Todd Kemp's earlier offering of a different course.