
# Math 142A: Introduction to Analysis I

## Lecture C (Charlesworth)

Last modified: December 22, 2017.

### Announcements

• Final exams are now visible on Gradescope. I will be holding special office hours for issues related to the grading of the final exam or the course during the first two weeks of next quarter, on Monday January 8th from 11:30 AM - 12:30 PM, and Tuesday January 16th from 1:30 PM - 3:15 PM. You must email me in advance carefully explaining what the issue is.

### Course Information

• Textbook: The textbook for this course is

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

This is the textbook available at the UC San Diego bookstore. Assignments will be posted fully on this website rather than referencing problems in the book, and so it should in theory be possible to complete this course using an entirely different text or no text at all; however, attempting to do so is not necessarily a wise choice. Errata for the textbook are available here.

• Exams: There will be two midterm exams and a final exam: midterms will be held during lectures on the dates given below, while the final has been scheduled and this information is available through the usual channels. Note that there will be no make-up dates for the midterms or the final exam.

• Piazza: This course will use Piazza, an online discussion board. It will allow you to post messages (openly or anonymously) and answer posts made by your fellow students, about course content, homework, exams, etc. The instructors will also monitor and post to Piazza regularly. This is the preferred method of contact for questions related to the course, especially if your question is likely to be relevant to other students as well. Note: the same Piazza group is shared with Professor Eggers' section of Math 142A; the courses will have very similar content, but not necessarily identical. Therefore, when asking a question about an assignment, it is particularly important to include the statement of the problem you are asking about.

### Homework

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.

Answers to these problems are now available here. The $\LaTeX$ file used to produce them is here.

1. #### The Tower of Hanoi

The Tower of Hanoi is a puzzle invented by Édouard Lucas in 1883. It consists of three pegs and several wooden discs of various sizes with holes cut in their centres so that they may be placed over the pegs. At the beginning of the puzzle, all of the discs are stacked on the leftmost peg, in order of decreasing size so that the smallest disc is at the top. The goal of the puzzle is to move all of the discs to the rightmost peg. However, only one disc may be moved at a time, and no disc may be placed on top of a disc which is smaller than it.

Prove that an optimal solution to a Tower of Hanoi with $n$ discs requires $2^n-1$ moves; that is, prove that no solution may use fewer moves, and that it is possible to solve it with this number of moves.

2. Let $S \subseteq \mathbb{R}$ be a subset of the real numbers. An element $a \in S$ is said to be a maximum of $S$ if it is an upper bound for $S$.

1. Prove that if $a\in S$ is a maximum of $S$, then $a = \sup S$.
2. Give an example of a non-empty set which is bounded above but has no maximum element.
3. Show that every non-empty finite set has a maximum element.
3. Determine whether each of the following statements is true or false, and provide a brief justification of your answer:
1. The empty set $\emptyset$ is bounded above by $5$.
2. Every element of the empty set $\emptyset$ is positive.
3. Every element of the empty set $\emptyset$ is negative.
4. If $S$ denotes the set $S := \{x \in \mathbb{R} | x^{-1} \in \mathbb{N}\}$ and $\alpha := \inf S$, then $\alpha > 0$.
5. Every real number is the least upper bound of some set of rational numbers.
6. For any real number $c$, the interval $\left(c - \frac{1}{4}, c + \frac{1}{4}\right)$ contains a rational number of the form $\frac{a}{3}$ with $a \in \mathbb{Z}$.
1. Show that there is a one-to-one and onto function from the integers $\mathbb{Z}$ to the natural numbers $\mathbb{N}$.
2. Show that there is a one-to-one function from the set of ordered pairs $\mathbb{N}^2 := \{(a, b) | a \in \mathbb{N}, b \in \mathbb{N}\}$ to the natural numbers $\mathbb{N}$. (Hint: you may use the fact that every natural number may be expressed as a product of prime numbers in a unique way; so, for example, if $2^s = 2^t$ then $s = t$.)
3. Show that there is a one-to-one function from the positive rational numbers $\mathbb{Q}_+ := \{\frac{a}{b} \in \mathbb{R} | a, b \in \mathbb{N}\}$ to the natural numbers $\mathbb{N}$. (Hint: you may use the previous part of this problem, and the fact that any positive rational number has a unique expression of the form $\frac{a}{b}$ where $a, b \in \mathbb{N}$ have no common factor.)

It turns out that there is no one-to-one function from the positive real numbers to the natural numbers. This fact was first demonstrated by Georg Cantor in the late 1800's. For more details, look up the Cantor diagonal argument.

4. Suppose that $f : A \to B$, and $Y \subset B$. We define the preimage of $Y$ to be the set $f^{-1}(Y) := \{x \in A | f(x) \in Y\}$. Likewise, recall that for $X \subset A$, the image of $X$ is the set $f(X) := \{f(x) \in B | x \in X\}$.
1. Show that for any $X \subset A$, $X \subseteq f^{-1}\left(f(X)\right)$.
2. Show that $f$ is one to one if and only if for every subset $X \subset A$, we have $f^{-1}\left(f(X)\right) = X$.
3. Show that $f$ is onto if and only if for every subset $Y \subset B$, we have $f\left(f^{-1}(Y)\right) = Y$.
Assignment 1, due October 6th, 2017.

Answers to these problems are now available here. In some case these are sketches rather than full proofs, but they should at least give an indication of how to prove the result.

1. Determine which of the following sequences in $n$ converge. Find the limits of those which do.
1. $\left(\frac{n}{n+1}\right)$
2. $\left(\sqrt{n+1}-\sqrt{n}\right)$
3. $\left(n\right)$
4. $(3)$
5. $\left(\frac{(-1)^n}{n^2}\right)$
2. Prove each of the following statements that is true. Disprove each that is false.
1. If the sequence $(a_n^2)$ converges, then $(a_n)$ also converges.
2. If the sequence $(a_n + b_n)$ converges, then $(a_n)$ and $(b_n)$ converge.
3. If $(a_n)$ converges to $a > 0$, then there is some $N \in \mathbb{N}$ such that $a_n > 0$ for every $n > N$.
3. Suppose that $(a_n)$ and $(b_n)$ converge to $a$ and $b$ respectively. Let $(c_n)$ be the sequence defined by $c_{2k} = a_k$ and $c_{2k-1} = b_k$. Show that $(c_n)$ converges if and only if $a = b$.
4. Let $(a_n)$ be a sequence. Show that $(a_n)$ converges to $a$ if and only if $(a-a_n)$ converges to $0$.
5. Let $S$ be a non-empty set which is bounded above, with least upper bound $m$. Show that there is a sequence $(a_n)$ such that $a_n \in S$ and $\displaystyle\lim_{n\to\infty}a_n = m$.
6. #### Bonus:

Does there exist a sequence $(r_n)$ such that for each $\alpha \in \mathbb{R}$ there is some subsequence $(r_{n_k})$ which converges to $\alpha$? Prove it or disprove it.
Assignment 2, due October 13th, 2017.

Answers to these problems are now available here. The $\LaTeX$ file used to produce them is here.

1. Let $(a_n)$ be a sequence of real numbers. Determine whether each statement below implies that $(a_n)$ converges. Then determine whether each statement below is implied by the assumption that $(a_n)$ converges.
1. There is some $a\in\mathbb{R}$ so that for every $\epsilon > 0$ there is some $N \in \mathbb{N}$ so that for every $n \geq N$, $|a_n-a| < 10\epsilon + \epsilon^2$.
2. For every $\epsilon > 0$ there is some $N \in \mathbb{N}$ so that for every $n > N$, $|a_{n+1}-a_n| < \epsilon$.
3. There is some $a \in \mathbb{R}$ and some $N \in \mathbb{N}$ so that for every $\epsilon > 0$ and every $n > N$, $|a_n-a| < \epsilon$.
4. The sequence $\left(|a_n|\right)$ is monotonic and bounded.
2. A set of real numbers $S \subseteq \mathbb{R}$ is said to be open if for every $x \in S$ there is some $\epsilon > 0$ such that $(x-\epsilon, x+\epsilon) \subset S$.
1. Show that for any real numbers $a < b$, the interval $(a, b)$ is open.
2. Show that if $G$ is open, then $\mathbb{R}\setminus G$ is closed.
3. Show that if $F$ is closed, then $\mathbb{R}\setminus F$ is open.
3. #### The Nested Interval Theorem

Suppose that $(a_n)$ and $(b_n)$ are sequences such that for every $n$, $a_{n} \leq a_{n+1} \leq b_{n+1} \leq b_n$, so that $[a_{n+1}, b_{n+1}] \subseteq [a_n, b_n]$. Prove that the set $S = \bigcap_{n=1}^{\infty} [a_n, b_n] = \left\{x \in \mathbb{R} : x \in [a_n, b_n]\,\forall n\right\}$ is non-empty.
4. Let $(b_n)$ be a bounded sequence of non-negative numbers. Show that for any $r \in [0, 1)$, the sequence of partial sums $(s_n)$ given by $s_n = \sum_{k=1}^n b_kr^k$ converges.
1. #### The Squeeze Theorem (for Sequences)

Show that if $(a_n), (b_n),$ and $(c_n)$ are sequences such that $(a_n)$ and $(c_n)$ converge to $L$, and for some $N \in \mathbb{N}$ and every $n > N$ we have $a_n \leq b_n \leq c_n$, then $(b_n)$ converges to $L$.
2. Show that if $(a_n)$ and $(b_n)$ are sequences such that $(a_n)$ diverges to infinity and for some $N \in \mathbb{N}$, every $n > N$, and some $c > 0$ we have $b_n > ca_n$, then $(b_n)$ diverges to infinity.
5. #### Bonus:

Suppose $(a_n)_n$ is a sequence with the property that every subsequence $(a_{n_k})_k$ has a further subsequence $(a_{n_{k_j}})_j$ which converges to $a$. Does $(a_n)_n$ converge to $a$? Prove it or provide a counterexample.
Assignment 3, due October 20th, 2017.

Answers to these problems are now available here. The $\LaTeX$ file used to produce them is here.

1. Suppose that $(a_n)$ is a bounded sequence.
1. Prove that for any $\epsilon > 0$ there are only finitely many indices $j$ for which $a_j > \displaystyle\limsup_{n\to\infty}a_n + \epsilon$.
2. Prove that for any $\epsilon > 0$ there are infinitely many indices $j$ for which $a_j > \displaystyle\limsup_{n\to\infty}a_n - \epsilon$.
3. Prove that $\displaystyle\limsup_{n\to\infty}a_n$ is the only real number with both of these properties.
4. Show that every bounded sequence has a subsequence which converges to its limit superior. (Note that this gives a second proof of the Bolzano-Weierstrass Theorem.)
2. Suppose that $S$ $\subseteq \mathbb{R}$ is a sequentially compact set: that is, every sequence in $S$ has a subsequence which converges to a point in $S$. Prove the following directly, without appealing to compactness from Section 2.5 of the book.
1. Show that $S$ is bounded.
2. Show that $S$ is closed.
3. Suppose that $g : \mathbb{R} \to \mathbb{R}$ is defined as follows: $$g : x \mapsto \begin{cases} x + 4a & \text{ if } x \leq -2 \\ ax + 2b & \text{ if } -2 < x \leq 3 \\ 2x^2 - b & \text{ if } 3 < x. \end{cases}$$ For what values of $a$ and $b$ is $g$ continuous?
4. Suppose that $f : \mathbb{R} \to \mathbb{R}$ has the property that for some $M \in \mathbb{R}$ and all $x \in \mathbb{R}$, $|f(x)| < M$. Show that $x \mapsto xf(x)$ is continuous at $0$.
5. Suppose that $f : [a, b] \to \mathbb{R}$ is continuous. Show that there is some $M \in \mathbb{R}$ such that $|f(x)| < M$ for all $x \in [a, b]$. (Hint: if not, show that there is a sequence of points $(x_n)$ so that $|f(x_n)| > n$...)
6. #### Bonus:

Let $K \subset \mathbb{R}$. An open cover of $K$ is a (possibly infinite) set $\mathcal{U}$ of open subsets of $\mathbb{R}$, such that $$K \subseteq \bigcup_{U \in \mathcal{U}} U.$$ That is, every point $x \in K$ is contained in at least one element of $\mathcal{U}$. A subcover of $\mathcal{U}$ is a subset $\mathcal{U}' \subseteq \mathcal{U}$ which is still an open cover of $K$.

The set $K$ is said to be compact if every open cover of $K$ has a finite subcover -- that is, if finitely many sets in any open cover suffice to cover $K$.

1. Show that if $K$ is compact, then $K$ is closed and bounded.
2. Show that the intersection of a closed set and a compact set is compact.
3. Show that the interval $[0, 1]$ is compact. (Hint: if it is not, there is some open cover $\mathcal{U}$ with no finite subcover; show that for such a cover, there is a nested sequence of shrinking closed subintervals, none of which have a finite cover by elements of $\mathcal{U}$, and look at their intersection.)
4. Conclude that if $K$ is closed and bounded, then $K$ is compact.

We have shown here that for subsets of $\mathbb{R}$, compact is equivalent to sequentially compact. It turns out that both of these notions may be extended to more complicated settings, in which they are no longer equivalent.

It turns out that this is the content of Section 2.5 of the text book. The new bonus problem is below. However, the above are still fun to try to work out without reading the book.
7. #### Bonus:

Consider the function $f : \mathbb{R} \to \mathbb{R}$ defined as follows: $$f : x \mapsto \begin{cases} 0 & \text{ if } x \notin \mathbb{Q} \\ 1 & \text{ if } x = 0 \\ \frac{1}{q} & \text{ if } x = \frac{p}{q} \text{ with } p > 0,\,p, q \in \mathbb{Z}, \text{ and } \mathrm{gcd}(p, q) = 1. \end{cases}$$ Show that $f$ is continuous at every irrational, and discontinuous at every rational.

#### Bonus bonus:

Is there a function which is continuous at every rational and discontinuous at every irrational? (This one is quite difficult.)

Assignment 4, due October 30th, 2017.

Answers to these problems are now available here. The $\LaTeX$ file used to produce them is here.

1. Prove that, at any given instant, there are two antipodal points on the equator with the same temperature. (You may assume that temperature varies continuously with position, and that the equator is a perfect circle; antipodal points are points on either end of a diameter of the circle.)
2. Let $h : \mathbb{R} \to \mathbb{R}$ be defined so that $h|_{[0, 1]} : x \mapsto x$, $h|_{[1, 2]} : x \mapsto 2-x$, and for all $x \in \mathbb{R}$, $h(x) = h(x+2)$. Note that $h$ behaves as a triangle wave, and that $h(x) = h(-x)$.

Define a function $f$ as follows: \begin{align*}f : \mathbb{R} &\longrightarrow \mathbb{R} \\ t &\longmapsto \begin{cases} h\left(\frac1x\right) & \text{ if } x \neq 0 \\ 0 & \text{ if } x = 0. \end{cases}\end{align*}

1. Show that $f$ is discontinuous at $0$, and continuous everywhere else.
2. Show that $f$ satisfies the conclusion of the intermediate value theorem: for every closed interval $[a, b]$ with $a \neq b$, and every $c$ between $f(a)$ and $f(b)$, there is some $x \in (a, b)$ so that $f(x) = c$.

What we've shown here is that not every function which satisfies the conclusion of the intermediate value theorem is continuous. In fact, things can be much, much worse: the Conway base 13 function takes on every real value in every non-empty open interval — in particular, it has the intermediate value property — but is horribly discontinuous at every point.

3. Show that if $f, g : X \to \mathbb{R}$ are continuous functions, then so is $f \vee g : t \mapsto \max(f(t), g(t))$. Show that the analogous statement about uniformly continuous functions is true.
4. Suppose $S \subseteq \mathbb{R}$ is not sequentially compact. Show that there is a continuous positive function $f : S \to (0, \infty)$ which does not achieve its minimum value. (Hint: you may wish to treat separately the case that $S$ is unbounded and the case that $S$ is not closed.)
5. Suppose that $f : (a, b) \to \mathbb{R}$ is a uniformly continuous function on the open bounded interval $(a, b)$. Show that $f$ is bounded.
6. #### Bonus:

The supremum norm of a function $f : X \to \mathbb{R}$ is defined to be the quantity $$\|f\|_\infty = \sup\{|f(x)| : x \in X\}$$ if the supremum exists, and infinity otherwise. A sequence of functions $(f_n)$, with $f_n : X \to \mathbb{R}$, is said to converge uniformly to a function $f : X \to \mathbb{R}$ if for every $\epsilon > 0$ there is $N \in \mathbb{N}$ such that for all $n > N$, $$\|f_n - f\|_\infty < \epsilon.$$ The sequence is said to converge to $f$ pointwise if for every $x \in X$, the sequence of real numbers $(f_n(x))$ converges to $f(x)$. Observe that uniform convergence implies pointwise convergence.

1. Prove that if $(f_n)$ is a sequence of continuous functions which converges uniformly to $f$, then $f$ is continuous.
2. Let $g_n : [0, 1] \to \mathbb{R}$ be defined by $g_n : x \mapsto x^n$. Show that $g_n$ converges pointwise to a discontinuous function.
3. #### Much more difficult:

Suppose that $(f_n)$ are a sequence of functions with $f_n : [a, b] \to [-M, M]$ such that for every $\epsilon > 0$ there is a $\delta > 0$ such that for every $n \in \mathbb{N}$ and $x, y \in [a, b]$, if $|x-y| < \delta$ then $|f_n(x) - f_n(y)| < \epsilon$. Show that there is a subsequence $(f_{n_k})$ which converges uniformly to some $f : [a, b] \to [-M, M]$.
7. (You don't need to turn this one in, but think about it for a couple minutes.) Does there exist an unbounded set $X \subseteq \mathbb{R}$ such that every continuous function $f : X \to \mathbb{R}$ is uniformly continuous?
Assignment 5, due November 8th, 2017.

Answers to these problems are now available here. The $\LaTeX$ file used to produce them is here.

1. For each of the following, evaluate the limit indicated or prove that it does not exist.
1. $\displaystyle\lim_{x \to 3} \frac{\sqrt{3x}-3}{x-3}$
2. $\displaystyle\lim_{x \to \infty} \frac{3x^4+6x^3-12x+3}{(2x+1)(5x^2+4x+16)(x-2)}$
3. $\displaystyle\lim_{x \to 0} \frac{3x^4 - 2x^3 + x^2 + 10x}{(4x^2-x)(x^5+x^3+x+2)}$
4. $\displaystyle\lim_{x\to0^+}\displaystyle\lim_{y \to 0} x^y$
5. $\displaystyle\lim_{y\to0}\displaystyle\lim_{x \to 0^+} x^y$
2. Suppose $f, g : S \to \mathbb{R}$ and $h : f(S) \to \mathbb{R}$ are monotonic functions. For each of the following functions, either prove it is monotonic or provide an example to show that it is not necessarily monotonic.
1. $f+g$
2. $fg$
3. $h\circ f$
4. $\frac1{f}$, assuming $0 \notin \color{red}{f(S)}$
1. Suppose that $S \subset \mathbb{R}$ is bounded above, $f : S \to \mathbb{R}$ is monotonic, and $x = \sup S$. Show that the limit from the left of $f$ at $x$ either converges, diverges to $\infty$, or diverges to $-\infty$.
2. As a consequence of the above, show that if $S \subset \mathbb{R}$ is arbitrary (i.e., potentially unbounded) and $x \in \mathbb{R}$ is the limit of some increasing sequence in $S$, then the limit from the left of $f$ at $x$ either converges, diverges to $\infty$, or diverges to $-\infty$.
3. Suppose that $f : \mathbb{R} \to \mathbb{R}$ is continuous. Show that $f$ is monotonic if and only if $f^{-1}(\{y\})$ is an interval (although potentially empty) for every $y \in \mathbb{R}$.
4. Evaluate the following limits:
1. $\displaystyle\lim_{x\to1}\frac{x^3-1}{x-1}$
2. $\displaystyle\lim_{x\to1}\frac{x^n-1}{x-1}$, for $n \in \mathbb{N}$
3. $\displaystyle\lim_{x\to1}\frac{x^n-x^m}{x-1}$, for $n, m \in \mathbb{N}$
5. #### Bonus:

A set $S$ is said to be countable if there is some map $f : S \to \mathbb{N}$ which is one-to-one. In the first assignment, you showed that $\mathbb{Q}_+$ and $\mathbb{N}^2$ were countable.

Suppose $f : \mathbb{R} \to \mathbb{R}$ is monotonically increasing. Show that the set of points at which $f$ is discontinuous is countable.

Assignment 6, due November 17th, 2017.

Answers to these problems are now available here. The $\LaTeX$ file used to produce them is here.

1. For each of the following functions, compute the derivative where it exists, and state the points at which it does not exist.
1. $\frac{x^2-1}{x^2+2x+1}$
2. $\frac{x^2-2x+1}{1-x^2}$
3. $\sin\left(\cos\left(2x^2\right)\right)$
4. $\frac{x-4}{\sqrt{|x|+1}}$
2. Let $a > 0$, and suppose that $f : \mathbb{R}\to\mathbb{R}$ is defined by $f(x) = a^x$. Show that if $f$ is differentiable at $0$, then $f$ is differentiable everywhere, and $f'(x) = a^xf'(0)$.
3. Let the function $f$ be defined as follows: \begin{align}f : \mathbb{R} &\longrightarrow \mathbb{R} \\ x &\longmapsto \begin{cases}x^2\sin\left(\frac1{x}\right)&\text{ if } x \neq 0\\0 & \text{ if } x = 0\end{cases}.\end{align} Show that $f$ is differentiable everywhere, but its derivative is discontinuous at $0$. (Hint: a lot of this problem is very similar to a problem on assignment 5.)

4. Suppose that $a < b$, and $g : [a, b] \to \mathbb{R}$ is continuous on its domain, and invertible. Note that, as a consequence, $g([a,b])$ is a closed interval.
1. Suppose that $g$ is differentiable at $x_0 \in (a, b)$, and that $g'(x_0) \neq 0$. Show that $g^{-1}$ is differentiable at $g(x_0)$, and find its derivative there.
2. Suppose that $g$ is differentiable at $x_0 \in (a, b)$, and that $g'(x_0) = 0$. Show that $g^{-1}$ is not differentiable at $g(x_0)$.
1. Suppose that $a < b$, and that $f : [a, b] \to \mathbb{R}$ is strictly increasing and differentiable at $x_0 \in (a, b)$. Show that $f'(x_0) \geq 0$.
2. Give an example of a strictly increasing differentiable function $f$ such that $f'(x) = 0$ for some $x$ in its domain.
5. #### Bonus:

Recall the definition of uniform convergence and of pointwise convergence from homework 5:
The supremum norm of a function $f : X \to \mathbb{R}$ is defined to be the quantity $$\|f\|_\infty = \sup\{|f(x)| : x \in X\}$$ if the supremum exists, and infinity otherwise. A sequence of functions $(f_n)$, with $f_n : X \to \mathbb{R}$, is said to converge uniformly to a function $f : X \to \mathbb{R}$ if for every $\epsilon > 0$ there is $N \in \mathbb{N}$ such that for all $n > N$, $$\|f_n - f\|_\infty < \epsilon.$$ The sequence is said to converge to $f$ pointwise if for every $x \in X$, the sequence of real numbers $(f_n(x))$ converges to $f(x)$.
One of the results from that assignment was that if a sequence of continuous functions converges uniformly, then its limit is a continuous function. In this problem we will demonstrate a function which is continuous everywhere, but differentiable nowhere.

For each $n \in \mathbb{N}$, let $f_n$ be the function defined as follows: \begin{align}f_n : \mathbb{R} &\longrightarrow \mathbb{R} \\ x &\longmapsto \sum_{j=0}^n \left(\frac{3}{4}\right)^{\color{red} j}\sin\left(9^{\color{red} j}\pi x\right).\end{align}

1. Show that the sequence $(f_n)$ converges pointwise to some function $f$.
2. Show that the convergence is uniform. (It follows, then, that $f$ must be continuous.)
3. Show that for any $x_0 \in \mathbb{R}$, there are points $y, z \in \mathbb{R}$ so that $|x_0 - y|, |x_0-z| \leq \frac1{9^n}$, but $|f(y) - f(z)| \geq \left(\frac34\right)^n$.
4. Conclude that $f$ is not differentiable at $x_0$.
Assignment 7, due December 1st, 2017.

Answers to these problems are now available here. The $\LaTeX$ file used to produce them is here.

1. Suppose that $f : [a, b] \to \mathbb{R}$ is continuous, $x_0 \in (a, b)$, and $f$ is differentiable on $(a, b) \setminus\{x_0\}$. Further suppose that $\displaystyle\lim_{x\to x_0} f'(x)$ exists and equals $L$. Show that $f$ is differentiable at $x_0$, and $f'(x_0) = L$.
2. Consider the function \begin{align}f : \R &\longrightarrow \R\\ x&\longmapsto \begin{cases}x + x^2\sin\paren{\frac1{x^2}} & \color{red}{\text{ if } x \neq 0} \\ \color{red}{0} & \color{red}{\text{ if } x = 0}.\end{cases}\end{align}
1. Confirm that $f$ is differentiable at $0$, and $f'(0) = 1$.
2. Prove that there is no interval of the form $[-\delta, \delta]$ with $\delta > 0$ on which $f$ is increasing.
(In fact, $g$ defined by $g(x) = x+x^2\sin\paren{\frac1x}$ also has this property, but it is a little harder to demonstrate it.)
1. Show that if $f$ is continuous on $[a,b]$ but not monotonic, there are points $a \leq x_1 \le x_2 \leq b$ so that $f(x_1) = f(x_2)$.
2. Suppose that $f : [a-\epsilon, b+\epsilon] \to \R$ is differentiable on $[a,b]$, and $f'(a) \color{red}\lt 0 \color{red}\lt f'(b)$. Show that $f$ is not strictly monotonic on $[a,b]$.
3. Imagine that $f : [a-\epsilon, b+\epsilon] \to \R$ is differentiable on $[a,b]$ and $f'(a) \color{red}\lt M \color{red}\lt f'(b)$. Show that there is some $c \in (a,b)$ so that $f'(c) = M$. (You may wish to consider the function $x \mapsto f(x) - Mx$.)
4. Show that there is no function $f : \R\to\R$ so that $$f'(x) = \begin{cases}1&\text{ if } x \in \Q \\ 0 & \text{ if } x \notin \Q.\end{cases}$$
5. #### Bonus:

Show that parts B and C become false if the red inequalities above are $\leq$ rather than $\lt$.
3. Suppose that $f : (-1, 1) \to \R$ has $n$ derivatives on $(-1,1)$.
1. Show that if the $n$-th derivative $f^{(n)}$ is bounded and $$f(0) = f'(0) = \cdots = f^{(n-1)}(0) = 0,$$ then $|f(x)| \leq M|x|^n$ for some $M > 0$ and all $x \in (-1,1)$.
2. Show that if $|f(x)| \leq M|x|^n$ for some $M > 0$ then $$f(0) = f'(0) = \cdots = f^{(n-1)}(0) = 0.$$
4. Show that if $f : (a,b)\to \mathbb{R}$ and $c \in (a,b)$, there is at most one function $F : (a, b) \to\R$ so that $F(c) = 0$ and $F'(x) = f(x)$.
5. #### Bonus:

Does there exist a function $f : \R \to \R$ which is smooth (i.e., infinitely differentiable at every point in its domain) and so that all the derivatives of $f$ vanish at $0$ (i.e., $f^{(n)}(0) = 0$ for all $n \geq 0$), but $f(1) \neq 0$?
Assignment 8, due December 8th, 2017.

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

 Name Role Office E-mail Ian Charlesworth Instructor AP&M 5880C ilc@math.ucsd.edu Yingjia Fu Teaching Assistant AP&M 5720 yif051@ucsd.edu Jack Geller Teaching Assistant AP&M 6452 jcgeller@ucsd.edu

My office hours will be held on Wednesdays from 3:00 PM - 4:00 PM (immediately after lecture), and on Tuesdays from 10:00 AM - 12:00 PM. The Wednesday office hour is dedicated to this class, while the time on Tuesday is open both to this course and the other course I am teaching this quarter.

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:

• begin the subject line with "Math 142A:"; this ensures that your message will not be caught in a spam filter;
• include your full name and student number; this ensures that we know who is contacting us; and
• state your issue clearly; this ensures that we know what you need help with.

### Class Meetings

 Date Time Location Lecture C00 (Charlesworth) Mondays, Wednesdays, Fridays 2:00pm - 2:50pm HSS 1330 Discussion C01 (Yingjia Fu) Thursdays 8:00am - 8:50am AP&M 5402 Discussion C02 (Yingjia Fu) Thursdays 9:00am - 9:50am AP&M 5402 Discussion C03 (Jack Geller) Thursdays 8:00am - 8:50am AP&M 7421 Final Exam Wednesday, Dec 13 3:00pm - 6:00pm HSS 1330

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### Syllabus

Course:  Math 142A

Title:  Introduction to Analysis I

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

Prerequisites:  Math 31CH or Math 109, or consent of instructor.

Catalog Description:  First course in an introductory two-quarter sequence on analysis. Topics include: the real number system, numerical sequences and series, limits of functions, continuity. See the UC San Diego Course Catalog.

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

Subject Material:  We will cover parts of chapters 1-4 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
0 Sep 25 Sep 26 Sep 27 Sep 28 Sep 29
Preliminaries, 1.1
1 Oct 2
1.1 The completeness axiom.
Oct 3 Oct 4
1.2 The distribution of $\mathbb{Z}$ and $\mathbb{Q}$.
Oct 5
Discussion
Oct 6
1.3 The triangle inequality; binomial coefficients; some finite series.
2 Oct 9
2.1 Sequences and convergence.
Oct 10 Oct 11
2.1 Convergence and sequences.
Oct 12
Discussion
Oct 13
2.2 Sequences and sets.
3 Oct 16
2.3 The monotone convergence theorem.
Oct 17 Oct 18
2.3, 2.4 Subsequences and the Peak Point Lemma.
Oct 19
Discussion
Oct 20
Catch up/Review.
4 Oct 23
Mid-term exam
Oct 24 Oct 25
2.4 The Bolzano-Weierstrass Theorem.
Oct 26
Discussion
Oct 27
3.1 Continuity.
5 Oct 30
3.2 The Extreme Value Theorem.
Oct 31 Nov 1
3.3 The Intermediate Value Theorem.
Nov 2
Discussion
Nov 3
3.4 Uniform continuity.
6 Nov 6
3.5 The $\epsilon-\delta$ criterion for continuity.
Nov 7 Nov 8
3.6 Monotone functions.
Nov 9
Discussion
Nov 10
Veterans Day
7 Nov 13
3.6 Monotone functions.
Nov 14 Nov 15
3.7 Limits.
Nov 16
Discussion
Nov 17
Catch up/Review.
8 Nov 20
Mid-term exam
Nov 21 Nov 22
4.1 Derivatives.
Nov 23
Thanksgiving
Nov 24
Post-Thanksgiving
9 Nov 27
4.1 Differentiation.
Nov 28 Nov 29
4.2 The Chain Rule.
Nov 30
Discussion
Dec 1
4.3 The Mean Value Theorem.
10 Dec 4
4.3, 4.4 The Cauchy Mean Value Theorem.
Dec 5 Dec 6
Catch up/Review.
Dec 7
Discussion
Dec 8
Review.
11 Dec 11 Dec 12 Dec 13 Dec 14 Dec 15

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. 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.

• It is your responsibility to ensure that you do not have a schedule conflict involving the final examination; you should not enroll in this class if you cannot take the final examination at its scheduled time.
• No calculators, phones, or other electronic devices will be allowed during the final examination. You may bring a single piece of paper of size no more than 8.5 inches by 11 inches containing hand-written notes, and at most three four-leaf clovers, horseshoes, maneki-neko, or other such talismans for good luck.
• There will be no makeup final exam.

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

Regrade Policy:

• Your exams will be graded using Gradescope.  You will be able to request a regrade directly from your TA for a specified window of time.  Be sure to make your request within the specified window of time; no regrade requests will be accepted after the deadline.

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

• It is your responsibility to check your scores and contact your TA before the end of the 10th week of the quarter to resolve recording errors.
• Questions regarding missing or incorrectly recorded scores will not be considered after the last day of instruction.

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
• Note: The above scale is a lower bound: for example, if your cumulative average is 80, your final grade will be at least B-. However, your instructor may adjust the above scale to be more generous.
Your cumulative average will be the best of the following two weighted averages:
• 20% Homework,  20% Midterm Exam I,  20% Midterm Exam II,  40% Final Exam
• 20% Homework,  20% Best Midterm Exam,  60% Final Exam

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.

• Entering/exiting class: Please arrive on time and stay for the entire class/section period. If, despite your best efforts, you arrive late, please enter quietly through the rear door and take a seat near where you entered. Similarly, in the rare event that you must leave early (e.g. for a medical appointment), please sit close to the rear exit and leave as unobtrusively as possible.
• Noise and common courtesy: When class/section begins, please stop your conversations. Wait until class/section is over before putting your materials away in your backpack, standing up, or talking to friends. Do not disturb others by engaging in disruptive behavior. Disruption interferes with the learning environment and impairs the ability of others to focus, participate, and engage.
• Electronic devices: Please do not use devices (such as cell phones, laptops, tablets, iPods) for non-class-related matters while in class/section. No visual or audio recording is allowed in class/section without prior permission of the instructor (whether by camera, cell phone, or other means).
• E-mail etiquette: You are expected to write as you would in any professional correspondence. E-mail communication should be courteous and respectful in manner and tone. Please do not send e-mails that are curt or demanding.

#### Accommodations:

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.

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### Resources

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

• UCSD Podcasts
• The lectures for this course will be podcast; you will be able to find the audio podcasts at the UCSD podcast site.
• Sites with resources for assistance and tutoring
• OASIS. The Office of Academic Support & Instructional Services (OASIS) offers weekly workshops to support many undergraduate classes. For more information, see the OASIS webpage.
• Calculus Tutoring  Links to information regarding the Calculus Tutoring Lab, OASIS, and private tutors
• Wolfram|Alpha  A powerful computational knowledge engine
• Sites with helpful advice for studying Mathematics
• Interesting sites relating to Mathematics (If you see a Math site you like that's not here, email me the URL.)

### Lecture Notes

Any remarks pertaining to particular lectures will be posted here throughout the term.
• Here is a typed version of the proof I skipped in class today, showing from the positivity axioms that $\{x \in \mathbb{R} | x^2 < 2\}$ is bounded above. If you are interested, the TeX file used to generate this is here.
CSS and page template greatfully taken from Todd Kemp's earlier offering of a different course.