CRN 12699

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  • Course: Math 3341 (Introduction to Analysis) - MW 16:00-17:20 in BELL 130A
  • Instructor: Dr. Helmut Knaust, Bell Hall 124, tel. 747-7002, e-mail:
  • Office Hours: M 17:30-18:20, R 11:00-12:00, or by appointment.
  • Textbook: Edward D. Gaughan, Introduction to Analysis. American Mathematical Society; 5th Revised edition. The textbook is currently on sale at the AMS website.
  • Prerequisites: The course requires a certain level of mathematical maturity that you have gained by having thoroughly and successfully grappled with the concept of infinity in MATH 1312 as well as the basics of logic and proofs in MATH 3325.
  • Course Objectives: Real Analysis is "Calculus with Proofs". You should expect (and I will expect) that you make considerable progress in the following areas:
  1. Become familiar with the fundamental results of "Analysis on the Real Line" (highlights of the course include the Intermediate Value Theorem, the Mean Value Theorem and possibly the Fundamental Theorem of Calculus);
  2. Thoroughly understand the definitions of the basic concepts of Analysis such as convergence, continuity, differentiation and integration;
  3. Continue to develop your ability to use the method of proof to establish these fundamental results.
  4. Be able to recognize a rigorous proof when you read or see one. Conversely, be able to pick out the weak spot(s) in a less rigorous argument. Be able to fill in details in a sketchy proof.
  5. Once you have devised a proof, be able to write it down in a clear, concise manner using correct English and mathematical grammar.
  6. Be able to present and defend a proof to a group of your peers.
  • Tests: Two exams will be given on Wednesday, September 29 and Monday, November 8. Each exam counts 20% of your grade.
  • In-class Activities and Presentations: Mathematics is not a spectator sport. I will regularly call on students to give presentations of problems. Your in-class work, evaluated for both quality and quantity, will account for 15% of your grade.
  • Homework. I will regularly assign written homework. The homework will be graded to account for 20% of your grade.
  • Final Examination: The final exam on Monday, December 6, 16:00-18:45, is comprehensive and mandatory. It counts 25% of your grade.
  • Time Requirement: I expect that you spend an absolute minimum of six hours a week outside of class on reading the textbook, preparing for the next class, reviewing your own notes, and completing assignments. Not surprisingly, it has been my experience that there is a strong correlation between class grade and study time.
  • Drop Policy. The class schedule lists Friday, October 29, as the last day to drop with an automatic "W". After the deadline, I can only drop you from the course with a grade of "F". Beginning with the fall 2007 semester, all freshmen enrolled for the first time at any Texas public college or university will be limited to six course withdrawals (drops) during their academic career. Drops include those initiated by students or faculty and withdrawals from courses at other institutions! This policy does not apply to courses dropped prior to census day or to complete withdrawals from the university.
  • Students with Disabilities. If you have a disability and need special accommodation, please contact the Disabled Student Services Office (DSSO) in Union East 106, 747-5148,

Open Problems



  • 10/11: Exercises 2.1, 2.2, 2.12.
  • 10/4: Investigate the limit behavior of \(f:(0,1]\to\mathbb{R}\), given by \(f(x)=\begin{cases}1 & (x \in \mathbb{Q})\\0 & (x \in \mathbb{R}\setminus \mathbb{Q})\end{cases}\)
  • 9/20: Exercises 1.27, 1.32ac.
  • 9/8: Exercises 1.14, 1.18.
  • 8/30: Exercises 1.1, 1.3.
  • 8/25: Prove Theorem 0.23. Exercise 0.44.

Written Homework

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