ADVANCED CALCULUS - University of Alabama at Birmingham

1 ADVANCED CALCULUSL ecture notes forMA 440/540 & 441/5412015/16 Rudi Weikard12345x-2-11logHxLBased on lecture notes by G. Stolz and G. WeinsteinVersion of September 3, 20161 ContentsFirst things The The The language of mathematics3 Chapter 1. The real Field Order The induction Counting and The least upper bound axiom11 Chapter 2. Sequences and Sums and the Series18 Chapter 3. A zoo of functions23 Chapter 4. Limits of Continuous The intermediate value theorem and some of its Uniform convergence and continuity29 Chapter 5. The mean value theorem and Taylor s Uniform convergence and differentiation36 Chapter 6.

2 Existence and uniqueness of Properties of The fundamental theorem of Integration of piecewise continuous Uniform convergence and integration43 Chapter 7. Special Generalized Trigonometric functions and their Analytic Sets of measure zero and some consequences49 Appendix A. Some set theory and Elements of Basics of set The recursion theorem54 Index55 First things The goalsOur goal in this class is threefold:(1) to obtain a body of knowledge in ADVANCED CALCULUS , the basis of the analysis ofreal-valued functions of one real variable;(2) to learn how to communicate ideas and facts in both a written and an oral form;(3) and, perhaps most importantly, to become acquainted with indeed, to master the process of creating conducting this class we shall try to model a mathematical community in whichboth collaboration and competition are prevalent.

3 This community is no, you are on the verge of discovering the foundations for a number of rules and recipes which havebeen successfully in use for some time. In the process you will recreate a body of knowledgealmost as if you were the first to discover it. However, as we have only nine months to dothis rather than a century or two, there will be some help available to you, most prominentlyin the form of these notes which will delineate broadly a path in which discovery will (orcould) this course it is allowed and, in fact, required to criticize the person on the board forflaws or incomplete arguments (you are a scientific community).

4 Criticism has to be leveledin a professional manner, in particular, it has to be free from any personal insults. At thesame time you have to learn to accept criticism without taking it personally. By learningto stand up for your ideas (or to accept that you made a mistake) you may get somethingout of this course which is of value not only in The rulesThe following rules, based on intellectual and academic honesty, will be in force.(1) Everybody will have the opportunity to present proofs of theorems. You will havethe proof written out on paper and present it with the help of a document camera.(2) The audience (including the instructor) may challenge a statement made in thecourse of the proof at any point.

5 (3) If the presenter is able to defend the challenged statement, he or she proceeds; ifnot, the presenter must sit down earning no points for this problem and losing theright to present again that day. The challenger may present his or her solution orelect to receive a challenge reward (see rules (10) and (12)).(4) A proof of a theorem will be considered correct if no one has objections (or furtherobjections). Its written version will then be published by uploading it to Canvas(it should have a title and the list of authors). The presenter and, if applicable,his or her collaborators (see rule (9)) will earn a total of 10 points at this THINGS FIRST(5) During class the instructor has the final decision on determining whether an ar-gument may stand or not.

6 His verdict may still be challenged after a proof is published (see rule (6)).(6) If someone other than an author discovers a flaw in a published proof, he or shewill get the opportunity to explain the mistake and present a correct proof for atotal of 20 points.(7) While presenting proofs you may only refer to those axioms and theorems in thenotes which occur before the one you are working on, to published proofs of suchtheorems, to the definitions, and to the appendix.(8) You must give credit where credit is due, , during your presentation you mustdeclare the points at which you had help and by whom.(9) It is also possible to report joint work.

7 In such a case 4 points will be earned forthe presentation while the other 6 are evenly distributed among the collaborators.(10) The successor of a presenter will be chosen as the student with the smallest numberof points among the volunteers taking into account the modification by rules (3),(11) and (12). A die is rolled, if necessary.(11) You may volunteer for a particular problem by an e-mail to me. This (in the orderreceived) establishes priority among volunteers with the same number of points.(12) For a student who has earned a challenge reward 20 points will be subtracted fromhis or her current score for the purpose of determining a successor.

8 At the timesuch a student is selected to prove a theorem the challenge reward expires.(13) Class attendance and participation is required. Absences from class are recordedin Canvas. After 10 absences from class 10 points will be subtracted from yourclass score and the count of absences is set again to zero.(14) There will be no partial credit except as described above to share Hints As there is no partial credit, be well prepared to answer whatever questions mayarise. There is no need to take notes, final proofs will be published . Think alonginstead. Try to earn points early. The index at the end of these notes may lead you to some of the necessary defini-tions (let me know of any omissions).

9 Often a proof will be fairly simple once one realizes that a previous result or aprevious method of proof can be used. Thus constant participation in class, evenwhen it s other people s turn to present, is highly advised! You should go over theproofs again at home and see if they are still clear to you, possibly rewriting themwith more details. Mathematical reasoning takes time. You may expect some frustration withoutit there would never be a sense of accomplishment. Plan to spend a lot of timethinking about a problem before writing down a final solution. Exercises are for private study. Points are only earned for proofs of theorems.

10 More difficult problems are marked with a . See me if you need help! THE LANGUAGE OF The language of mathematicsMathematics is a highly formalized subject. In many ways learning it is similar to learn-ing a foreign language. This language rests on two (tightly interwoven) pillars: logic and settheory. In this course we assume that you have a basic familiarity (possibly unconsciously)with both. Nevertheless Appendix A collects some fundamental material about set theorywhich you may (and will have to) use in your proofs. If you like you may also find someinformation in my algebra notes which are on my website1. There, at the ends of and , are references to books on these (true or false) are the bread and butter of mathematics.