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Lecture Notes, Construct Examples and Counterexamples, Draw Pictures Analysis II Limits, continuity, differentiability, integrability Christian P. Jäh February 12, 2019 Foreword These note have been written primarily for you, the student. I have tried to make it easy to read and easy to follow. I do not wish to imply, however, that you will be able to read this text as it were a novel. If you wish to derive any benefit from it, read each page slowly and carefully. Youmust have a pencil and plenty of paper beside you so that you yourself can reproduce each step and equation in an argument. When I say verify a statement, make a substitution, etc. pp., you yourself must actually perform these operations. If you carry out the ex- plicit and detailed instructions I have given you in remarks, the text, and proofs, I can almost guarantee that you will, with relative ease, reach the conclusions. One final suggestion. As you come across formulas, record them and their equations/page numbers on a separate sheet of paper for easy reference. You may also find it advan- tageous to do the same for Definitions and Theorems. These wise words are borrowed from Morris Tenenbaum and Harry Pollard from the be- ginning of their book Ordinary differential equations. I could not have said it better and it certainly applies to this course. i Figure 0.1: Don’t just read it; fight it. – Paul Halmos (The comic is abstrusegoose.com) Contents How to study for Analysis II i Polya’s algorithm for solving problems iv List of symbols v List of (named) theorems viii List of important definitions x Glossary xii 1 Sequences and series in Rd 1 1.1 Some notational remarks ....................... 1 2 1.2 Convergence of sequences (an) ⊆ R ................ 2 1.3 Length and distance in Rd ....................... 8 1.3.1 The notion of a norm ...................... 8 1.3.2 The notion of a metric ..................... 17 1.4 Conclusions .............................. 21 1.5 The Bolzano–Weierstrass theorem ................... 29 2 Open, closed, and compact 30 2.1 Open balls in Rd ............................ 30 2.2 Open sets ................................ 32 CONTENTS i 2.3 Closed sets ............................... 36 2.4 Compactness ............................. 51 2.4.1 Properties of compact sets ................... 56 3 Limits of functions 61 3.1 Some priming ............................. 63 3.1.1 An abstract point of view .................... 64 3.2 Limits of Rm-valued functions .................... 65 3.3 Examples ................................ 67 3.4 Characterization of limits via sequences . 72 3.5 Arithmetic rules for limits ....................... 76 3.6 One sided limits of functions on R ................... 77 4 Continuity 79 4.1 We start by thinking about non-continuous functions ......... 79 4.2 Definition of continuity ......................... 82 4.3 Examples ................................ 89 4.4 Discontinuity .............................. 95 4.4.1 Classification of discontinuities . 98 4.4.2 Further counterexamples in continuity . 98 4.5 Continuity and component-wise continuity . 100 4.6 Operations with continuous functions . 102 4.7 Continuous functions f :[a; b] ! R . 107 4.7.1 Two important properties of continuous functions . 108 4.7.2 The Intermediate Value Theorem (IVT) . 112 4.7.3 Applications of the IVT . 115 4.7.4 Weierstrass’ Extremal Value Theorem . 118 4.8 Continuity of linear maps . 123 5 Differentiability and derivative on R1 125 5.1 Historical comments on Differential Calculus . 125 5.2 Some thinking ............................. 127 CONTENTS ii 5.3 Definition ................................ 128 5.3.1 Examples of derivatives . 132 5.3.2 Differentiability ) Continuity . 137 5.3.3 Are derivatives continuous? . 138 5.4 Operations with differentiable functions . 142 5.5 Properties of differentiable functions . 149 5.6 Derivatives of higher order . 155 5.7 Function spaces ............................ 155 6 Function sequences 160 6.1 Notions of convergence . 160 6.2 Examples & counterexamples . 168 6.3 (Real) Power series . 169 7 Integration on R1 172 7.1 Step functions and their integrals . 172 7.1.1 Properties of the step function integral I(f) . 183 7.2 Regulated functions and their integrals . 186 7.2.1 Higher dimensions . 199 7.3 The Riemann-Integral . 200 7.3.1 Properties of the Riemann Integral . 208 7.4 The fundamental theorem of calculus and primitives . 209 7.5 Some Rules for Integration . 214 7.6 Uniform convergence and integration . 219 8 Improper Integrals on on R1 225 8.1 Appendix : The closure of sets . 231 9 Differentiability and derivative on Rd 232 9.1 Definition ................................ 232 9.1.1 Directional Derivative . 238 9.1.2 Partial Derivatives . 239 9.2 Meaning of Differentiability in Rd . 243 CONTENTS iii A Prerequisites 246 A.1 Some notation used in this notes . 246 A.1.1 Operations on sets . 249 A.1.2 Some properties of sets . 251 A.2 Some Linear Algebra (of real vector spaces) . 253 A.2.1 Basis and dimension . 258 A.2.2 Linear maps . 260 A.3 Sequences ............................... 261 A.4 Series ................................. 268 A.5 Elementary properties of functions of one variable . 271 A.5.1 Restrictions of functions .................... 272 A.5.2 Monotonic functions ...................... 273 A.5.3 Odd and even functions .................... 273 A.5.4 Convex and concave functions ................. 274 A.6 Elementary inequalities ........................ 276 A.7 Cauchy–Schwarz, Minkowski, and Hölder ................ 278 Bibliography 282 How to study for Analysis II • Read carefully and deliberately. As you know, the way one should read in math- ematics is quite different from how you may read a history book or a magazine or newspaper. In mathematics you must read slowly, absorbing each phrase. In the first semester, you should all have received a self-explanation training book- let which you should not forget. If you have lost your copy or have never received one, you can contact me and I will bring some to the lecture. • Think with pencil and scratch paper. Working on mathematics you should al- ways have a pencil and some sheets of paper ready and use them. Test out what is written in the lecture notes, construct examples and counterexamples, draw pictures. This will help to clinch the ideas and procedures in your mind before starting the exercises. After you have read and reread a problem carefully, if you still do not see what to do, do not just sit and look at it. Get your pencil going on scratch paper and try to dig it out. Try this ”algorithm”: 1. Write down what you want to show. 2. Write down what you know. 3. What can you immediately deduce from the known? Apply some known simple inequalities and identities and see what you can get from it. 4. Think of a strategy. And try to implement it. iv CONTENTS v 5. Have you used all information? Do you have numbers, vectors or functions. Have you used their specific properties? Another good reason for getting things down is that it is much more easy to help you. Maybe you just overlooked something and with a small hint from me or tu- tors, you have a light-bulb experience. You should not rob yourself of that by ask- ing to much of the solutions at once. • Be independent. To be clear, being independent does not mean that you should ask no questions at all but to have good judgement over what to ask and when. Sometimes little things will cause considerable confusion or you do not know where even to begin with your studies. Then you should ask for help. Do not be afraid that your question may sound dumb. The only dumb action is to fail to ask about a topic that you have really tried to grasp and still do not understand. Some people seek help too soon and some wait too long. You will have to use good common sense in this matter. • Persevere. Do not become frustrated if a topic or problem may completely baf- fle you at first. Stick with it! An extremely interesting characteristic of learning mathematics is that at one moment the learner may feel totally at a loss, and then suddenly have a burst of insight that enables her to understand the situation per- fectly. If you don’t seem to be making any progress after working on a problem for some time, put it aside and attack it again later. Many times you will then see the solution immediately even though you have not been consciously thinking about the problem in the meantime. There is a tremendous sense of satisfaction in having been persistent enough and creative enough to independently solve a problem that had given you a great deal of trouble. • Take time to reflect To learn mathematics well you must take time to do some reflective thinking about the material covered during the last few days orweeks. It takes time for some ideas in mathematics to soak in. You may have to live with them a while and do reflective thinking about them before they become a partof you. CONTENTS vi • Concentrate on fundamentals. Do not try to learn mathematics by memorizing illustrative examples.1 You will soon become overwhelmed by this approach, and the further you go the less successful you will be. All mathematics is based on a few fundamental principles and definitions. Some of these must be memorized. But if you concentrate on these fundamentals and try to see how each new topic is just a reapplication of them, very little additional memorization will be neces- sary.2 • Use Heuristics.3 If you work on a problem that you can not solve immediately try to consider a problem that is similar but somewhat easier. It is not always easy to find such problems but it will come to you with practice.
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