Date  Event  
Monday, Jan. 13  First day of class  
Friday, Jan. 17  Last day to add a course  
Monday, Jan. 20  Martin Luther King holiday (all classes canceled)  
Monday, Jan. 27  Last day to drop a course  
HW #1  Friday, Jan. 24 
Exam study problems (these problems need not be turned in):
Section 1.1: 1, 2, 5, 6, 7. Define injective, surjective, and bijective. Graded problems (these problems should be turned in): Section 1.1: 3, 4 Section 1.2: No problems. (Problem 5 was assigned, but it seems clear upon rereading that there is an error in Problem 5.) Homework is to be turned in in gradable condition. In particular:

HW #2  Friday, Jan. 31 
Exam study problems (these problems need not be turned in):
Section 1.2: 2, 4, 7, 9, 10, 11, 12, 15. Define a Dedekind cut. Define supremum and infimum. For real numbers (Dedekind cuts) α and β, define α≤β, α+β, α, αβ when α>0 and β>0, and αβ when α and β are arbitrary. Graded problems (these problems should be turned in): (AB 1): Suppose that α, β, and γ are real numbers with β<γ. (a) Prove that α+β≤α+γ. (b) Prove that α+β≠α+γ. Section 1.2: 13, 14 
HW #3  Friday, Feb. 7 
Homework 1 rewrite due
Exam study problems (these problems need not be turned in): Section 1.3: 1, 2, 8, 9, 12, 13, 14, 15, 16. State the εN condition for convergence. Define lim_{n} a_{n}=∞ and lim_{n} a_{n}=∞. Define lim inf and lim sup. Graded problems (these problems should be turned in): Section 1.3: 10, 11b, 17 
Friday, Feb. 14 
Homework 2 rewrite due
First midterm exam—good luck! You will be allowed a nongraphing calculator and a doublesided 3 inch by 5 inch card of notes.  
HW #4  Friday, Feb. 21 
Homework 3 rewrite due
Exam study problems (these problems need not be turned in): Section 1.4: 1, 3, 4, 5, 6, 8, 9. Define what it means for a series to converge to the real number a. Define what it means for a series to converge absolutely. State the Comparison Test, the Ratio Test, and the Root Test. (AB 1.5): Prove the alternating series test: show that if {a_{n}} is a decreasing sequence of positive numbers, then ∑^{∞}_{n=1}(1)^{n}a_{n} is a convergent series. Section 1.5: Define finite, countable, countably infinite, and uncountable sets. Graded problems (these problems should be turned in): Section 1.4: 2, 7. Problem 2b should read ∑^{∞}_{n=1} x a_{n} and not x∑^{∞}_{n=1} a_{n}. Section 1.5: 1 
HW #5  Friday, Feb. 28 
Exam study problems (these problems need not be turned in):
Section 1.5: 2 Section 1.6: 1, 3, 4, Define a closed set. Define an open set. State an important equivalent condition for openness. Define the diameter of a set E⊆ℝ. Graded problems (these problems should be turned in): Section 1.5: Problem 3 is written incorrectly (other conditions must be imposed and other possibilities exist). Rewrite Problem 3 so that it is correct, and then solve your corrected problem. Section 1.6: 5, 6 
HW #6  Friday, Mar. 6 
Homework 4 rewrite due
Exam study problems (these problems need not be turned in): Section 1.6: 8, 9 (there is a typo in Problem 9), 10, 11. Define dist(x,E) for x∈ℝ and E⊆ℝ. Define the closure of a set E⊆ℝ. Section 1.7: 2, 4, 5, 6, 7 (do not use Proposition 1.7.11), 9, 10, 11, 12, 13. Why is the condition that X_{k} be closed necessary? 15, 16, 18, 20. Define a continuous function. State the εδ condition for a function to be continuous at a point. State the Extreme Value Theorem and the Intermediate Value Theorem. Define f ∨g and f ∧g. Define uniformly continuous function. Define Lipschitz function. (AB 2): Let X⊆ℝ. Show that f :X→ℝ is uniformly continuous if and only if, whenever x∈ℝ, {x_{n}}⊆X and {y_{n}}⊆X with x_{n}→x and y_{n}→x, we have that f (x_{n})f (y_{n})→0. (This problem replaces Problem 1.7.19, which I believe is stated incorrectly.) Graded problems (these problems should be turned in): Section 1.6: 7 Section 1.7: 3, 17. On Problem 3, justify your answer. A graph of the function in Problem 1.7.3 may be found here. 
Monday, Mar. 16, 9:00 a.m.  Homework 5 rewrite due
Due to our transition to online classes, homework should be submitted online. Bonus points for making my life easier are available on the Homework 5 rewrite and all subsequent original assignments (but not rewrites) as follows:
 
Friday, Mar. 20, 10:45 a.m.–12:45 p.m.  Exam 2 conducted online. Exam 2 will be made available in the Blackboard content tab at 10:45 a.m. You are to download the exam, write your answers on blank paper (or, at your discretion, type your answer), and scan and upload your answers by 12:45 p.m. Answers may be uploaded to Blackboard or emailed. Blackboard submission is preferred; if you can't figure Blackboard out, please email me the submission with a subject line of “Math 4513 Exam 2”.  
Mar. 23–27  Spring Break—have fun! Office hours are by appointment only during Spring Break.  
Review of online course structure
 
HW #7  Wednesday, Apr. 1, 9:00 a.m. 
Homework 6 rewrite due
Exam study problems (these problems need not be turned in): (AB 3): Show that if I⊆ℝ is an interval and f :I→ℝ is strictly increasing and continuous, then f ^{ −1} is also continuous. You may use the fact (proven in class) that this is true if in addition I is open. (AB 4): Suppose that X⊆ℝ (but need not be an interval) and that f :X→ℝ is strictly increasing and continuous. Is it necessarily the case that f ^{ −1} is continuous? Section 2.1: 1, 2, 3, 5, 7 Section 2.2: 1, 2, 4 (AB 5): (abcd) Problem 2.1.8 as stated is impossible: if lim_{x→0} f (x)=∞ and lim_{x→0} g(x)=−∞ then lim_{x→0} f (x)g(x)=−∞. The problem is possible if we replace f (x)g(x) by f (x)+g(x). (This is a graded problem; see below.) It is also possible if we retain f (x)g(x) and impose the condition lim_{x→0} f (x)=0 rather than lim_{x→0} f (x)=∞. Solve Problem 2.1.8(abcd) under this assumption on f . (e) Find an example of functions f and g such that lim_{x→0} f (x)=0, lim_{x→0} g(x)=−∞, and lim_{x→0} f (x)g(x)=−3. (f) Can you find an example of functions f and g such that lim_{x→0} f (x)=0, lim_{x→0} g(x)=−∞, and lim_{x→0} f (x)g(x)=+3? (AB 6): What conditions can we impose on lim_{x→0} f (x) and lim_{x→0} g(x) so that lim_{x→0} f (x)/g(x) can be −∞, −3, 0, 3, ∞ and does not exist? Graded problems: (These problems should be turned in via Blackboard. A LaTeX template for typing your homework is here and a quick guide to using it is here.) Section 2.1: 4, 8. In Problem 8, it should be f (x)+g(x) and not f (x)g(x) throughout. Section 2.2: 3 
HW #8  Monday, Apr. 6, 9:00 a.m. 
Exam study problems (these problems need not be turned in):
Section 2.2: 7, 8, 12 Section 2.3: 1, 2, 6, 9. In 2.3.9 you may assume that the trigonometric functions can be rigorously defined and that all the basic properties of trigonometric functions you know can be proven. Graded problems: (These problems should be turned in via Blackboard. A LaTeX template for typing your homework is here and a quick guide to using it is here.) Section 2.2: 10 Section 2.3: 3, 8 Bonus, 5 points (available on HW #8 or HW #9 but not both): Log on to Blackboard Collaborate at least once during office hours or our regularly scheduled class time and get your video sharing to work. 
HW #9  Monday, Apr. 13, 9:00 a.m. 
Homework 7 rewrite due
Exam study problems (these problems need not be turned in): Section 2.4: 3, 4ab, 6, 7, 8, 9 Section 2.5: 1 (first part involving Theorem 2.5.1), 2, 4 Graded problems: (These problems should be turned in via Blackboard. A LaTeX template for typing your homework is here and a quick guide to using it is here.) Section 2.4: 2 Section 2.5: 1 (second part only; that is, either show that Theorem 2.5.2 is valid with L=∞ or provide a counterexample). (AB 7) Let a<c<b. Suppose that f ∈C^{(n)}(a,b) and f ^{(n)} is differentiable on (a,b)\{c}. Let P_{c}(x)=∑_{k=0}^{n} f ^{(k)}(c) (xc)^{k}/k! be the Taylor polynomial for f of degree n expanded about c. Suppose that G is continuous on (a,b), and that G is differentiable on (c,b) with G'(z)>0 for all z∈(c,b). (a) Show that for every x in (c,b) there is a y with c<y<x and with f (x)=P_{c}(x)+f ^{(n+1)}(y) (xy)^{n} (G(x)G(c))/n!G'(y). (b) Find a value of G(y) that yields the formula f (x)=P_{c}(x)+f ^{(n+1)}(y) (xc)^{n+1}/(n+1)! proven in class. (There is more than one correct answer.) 
Friday, Apr. 17, 10:45 a.m. 
Exam 3 conducted online.
PDFs containing the exam problems will appear in the Blackboard Content tab at 10:45 a.m. on Friday, April 17. Please write your answers (by hand; do not type your answers) on paper, a whiteboard, or a tablet, then scan, photograph or export your answers and upload to Blackboard by 12:00 p.m. (noon). The exam is open book, open note, and open calculator (and you will need your book); however, you are not allowed to discuss the exam in any way with any person other than Professor Barton while taking it. 

Friday, Apr. 17  Last day to withdraw from a course.
If you have three final exams scheduled on the same day, then under university policy you are entitled to an alternative exam date; please inform me by email (aeb019@uark.edu) that you need a makeup exam on or before April 17.  
Monday, Apr. 20, 9:00 a.m.  Homework 8 rewrite due  
HW #10  Monday, Apr. 27, 9:00 a.m. 
Homework 9 rewrite due
Exam study problems (these problems need not be turned in): Section 3.1: 2, 5 (show that ∫_{a}^{b}f =∫_{a}^{b}g), 6 Graded problems: (These problems should be turned in via Blackboard. A LaTeX template for typing your homework is here and a quick guide to using it is here.) Section 3.1: 1, 4, 7. On (4) you may assume f is increasing, as the case where f is decreasing is similar. On (7) you may assume that a<b as the conclusion is not true if a=b. 
Wednesday, Apr. 29  Last day of class  
HW #11  Thursday, Apr. 30, 11:59 p.m. 
Exam study problems (these problems need not be turned in):
Section 3.1: 8, 9, 10 (do not assume a<b<c). Section 3.2: 2, 4, 6 Section 3.3: 1, 2, 4, 5, 6, 7, 10, 11, 12 Graded problems: (These problems should be turned in via Blackboard. A LaTeX template for typing your homework is here and a quick guide to using it is here.) Section 3.2: 1 (do only the left endpoint a), 7 Section 3.3: 3 
Friday, May 1  Complete the online course evaluation on or before this date. If at least 70% of the class completes the course evaluation before the deadline, I will drop your 2 lowest homework scores; otherwise, I will drop your 1 lowest homework score.  
Monday, May 4 
Homework 10–11 rewrites due Final exam, 10:15 a.m.–12:15 p.m. or time indicated on the registrar's website. Good luck! 
Instructor  Ariel Barton (email aeb019@uark.edu) 
Lecture  Video lectures hosted through Blackboard; discussion/Q&A sessions will be hosted through Blackboard Collaborate Monday, Wednesday, Friday, 10:45–11:35 a.m. 
Office hours  Via Blackboard Collaborate, MWF 10:45–11:35 a.m. or by appointment. 
Course Description: The real and complex number systems, basic set theory and topology, sequences and series, continuity, differentiation, and Taylor's theorem. Emphasis is placed on careful mathematical reasoning.
Prerequisites: MATH 2574, MATH 2803, and MATH 3083 or MATH 3093.
Text: A First Course in Analysis, John B. Conway, ISBN 9781107173149. A paper copy may be bought from the bookstore; electronic copies may be purchased from several online retailers, including Amazon.com and Barnes and Noble.
Course grade: Here is how I plan to weigh your grades:
Homework  24% 
Highest scoring midterm test  19% 
Median scoring midterm test  19% 
Lowest scoring midterm test  0% 
Final  38% 
Homework: Assignments will be posted to this website. You may ask anyone for help with your homework, but you must write up your solutions on your own. Assignments should be turned in for grading via Blackboard by the due date.
Late homework will not be accepted except in the case of a medical or other unforeseeable emergency, and documentation will be required.
I expect to have 11 assignments over the course of the semester; your lowest score (or two scores) will be dropped.
Tests: There will be three midterms tests and a final exam. I plan to hold midterm exams during class time on the dates indicated in the calendar above. The final exam is currently expected to occur at the time indicated on the registrar's website.
Students may use nongraphing calculators and portable timepieces on exams. All other electronic devices, including watches that do anything other than tell the time and date, are prohibited.
If you require accommodations on an exam, notify your instructor as soon as possible, but in all cases at least one week before the exam is to be held. Documentation from the CEA may be required, depending on the nature of the accommodation.
Makeup exams: Makeup exam requests also require written documentation as to your conflict. Except in the case of medical or other unforeseen emergencies, makeup exam requests must be made at least one week before the exam is to be held. Makeup exams are at the instructor's discretion; if you do not provide a documented reason why you cannot take the exam at the usual time, if your reason is considered inadequate, or if your request for a makeup exam is not made in a timely fashion, I reserve the right to refuse a makeup exam or to assess a late penalty (deduction from your score).
If you have three or more final exams scheduled for Monday, May 4, then under University policy you are entitled to reschedule one of your finals. If you wish to reschedule the final for our class, please notify me by email by April 17, and I will arrange for you to take a makeup final later in the week.
Incompletes: Only given in extreme circumstances, and only when the student has satisfactorily completed all but a small portion of the work in the course. Students must make prior arrangements with the professor well before the end of the semester.
Academic Integrity: Academic dishonesty on any exam, quiz, or other graded item will result in a score of zero that cannot be dropped or replaced. Suspected cases of academic dishonesty are referred to the AllUniversity Academic Integrity Board. The following passage is quoted from the referenced website and is the policy in this course:
As a core part of its mission, the University of Arkansas provides students with the opportunity to further their educational goals through programs of study and research in an environment that promotes freedom of inquiry and academic responsibility. Accomplishing this mission is only possible when intellectual honesty and individual integrity prevail. Each University of Arkansas student is required to be familiar with and abide by the University's Academic Integrity Policy which may be found here. There are harsh penalties for violations as prescribed by the Sanction Rubric. Students with questions about how these policies apply to a particular course or assignment should immediately contact their instructor.
Commercial Note Vendors: Some commercial vendors may reach out to you to sell the notes you take in this class. Notes derived from class lectures are the intellectual property of the instructor. Selling or otherwise sharing these notes outside the class is a violation of the instructor's intellectual property rights and constitute a violation of the University's academic integrity policies. Your continued enrollment in this class signifies your understanding of and your intent to abide by this policy.