具体描述
The transition from studying calculus in schools to studying mathematical analysis at university is notoriously difficult. In this third edition of Numbers and Functions, Professor Burn invites the student reader to tackle each of the key concepts in turn, progressing from experience through a structured sequence of more than 800 problems to concepts, definitions and proofs of classical real analysis. The sequence of problems, of which most are supplied with brief answers, draws students into constructing definitions and theorems for themselves. This natural development is informed and complemented by historical insight. Carefully corrected and updated throughout, this new edition also includes extra questions on integration and an introduction to convergence. The novel approach to rigorous analysis offered here is designed to enable students to grow in confidence and skill and thus overcome the traditional difficulties.
作者简介
目录信息
Preface to second edition xxi
Glossary xxii
PART I NUMBERS
1 Mathematical induction 3 Mathematical induction (qns 1—8) 3
Historical note 6 Answers and comments 8
2 Inequalities 10 Positive numbers and their properties (qns 1—29) 10
Summary — properties of order 14
Arithmetic mean and geometric mean (qns 30—39) 14
Completing the square (qns 40—42) 16
The sequence (1 Y 1/n)v (qns 43—49) 17
nth roots (qns 50, 51) 18
Summary — results on inequalities 18
Absolute value (qns52—64) 19
Summary — results on absolute value 21
Historical note 21
Answers and comments 22
3 Sequences: a first bite at infinity 28
Monotonic sequences (qn 4) 29
Bounded sequences (qns 5—7) 30
Subsequences (qns 8—16) 31
Sequences tending to infinity (qns 17, 18) 35
Archimedean order and the integer function (qns 19—23) 36
Summary — the language of sequences 37
Null sequences (qns 24—47) 38
Summary — null sequences 44
Convergent sequences and their limits (qns 48—60) 45
Boundedness of convergent sequences (qns 61—63) 49
Quotients of convergent sequences (qns 64—69) 49
d’Alembert’s ratio test (qns 70—74) 50
Convergent sequences in closed intervals (qns 75—80) 52
Intuition and convergence (qns 81—83) 53
Summary — convergent sequences 55
Historical note 57
Answers and comments 59
4 Completeness: what the rational numbers lack 68
The Fundamental Theorem of Arithmetic (qns 1—3) 68
Dense sets of rational numbers on the number line (qns 4—10) 69
Infinite decimals (qns11—17) 70
Irrational numbers (qns 18—21) 72
Infinity: countability (qns 22—30) 73 Summary 75
The completeness principle: infinite decimals are convergent (qns 31, 32) 76
Bounded monotonic sequences (qns 33—36) 78
nth roots of positive numbers, n a positive integar (qns37—41) 79
Nested closed intervals (qn 42) 80
Convergent subsequences of bonded sequences (qns 43—47) 81
Cluster points (the Bolzano—Weierstrass theorem) (qns 48—54) 81
Cauchy sequences (qns 55—58) 83
Least upper bounds (sup) and greatest lower bounds (inf) (qns 59—82) 85
Upper bounds and greatest terms (qns 59—61) 85
Least upper bound (sup) (qns 62—66) 86
Lower bounds and least members (qns 67—70) 87
Greatest lower bound (inf) (qns 71—78) 88
sup, inf and completeness (qns 79—82) 89
lim sup and lim inf (qns 83, 84) 90
Summary — completeness 91
Historical note 92
Answers and comments 95
5 Series: infinite sums 104
Sequences of partial sums (qns 1—10) 104
The null sequence test (qns 11—13) 107
Simple consequences of convergence (qns 14—22) 107
Summary — convergence of series 108
Series of positive terms 109
First comparison test (qns 23—29) 109
The harmonic series (qn 30) 110
The convergence of Σ1/n^α (qns 31, 32) 110
Cauchy’s nth root test (qns 33—39) 111
d’Alembert’s ratio test (qns 40—50) 112
Second comparison test (qns 51—55) 113
Integral test (qns 56—61) 114
Summary — series of positive terms 116
Series with positive and negative terms 117
Alternating series test (qns 62—65) 117
Absolute convergence (qns 66 —70) 118
Conditional convergence (qn 71) 119
Rearrangements (qns 72—77) 120
Summary — series of positive and negative terms 122
Power series 123
Application of d’Alembert’s ratio test and Cauchy’s nth
root test for absolute convergence (qns 78—90) 123
Radius of convergence (qns 91—101) 123
Cauchy—Hadamard formula (qns 102—107) 125
The Cauchy product(qns108—113) 126
Summary — power series and the Cauchy product 128
Historical note 129
Answers and comments 131
PART II FUNCTIONS
6 Functionsandcontinuity:neighbourhoods,limits of functions 143
Functions (qn 1) 143
The domain of a function 143
The range and co-domain of a function(qns2—4) 144
Bijections and inverse functions (qns 5—6) 145
Summary — functions 145
Continuity (qns 7—11)
Definition of continuity by sequences (qns 12—18) 147
Examples of discontinuity (qns 19, 20) 148
Sums and products of continuous functions (qns 21—31) 149
Continuity in less familiar settings (qns 32—35) 150
A squeeze rule (qns 36, 37) 151
Continuity of composite functions and quotients of
continuous functions (qns 38—55) 151
Summary — continuity by sequences 154
Neighbourhoods (qns 56—63) 154
Definition of continuity by neighbourhoods (qns 64—72) 156
One-sided limits 159
Definition of one-sided limits by sequences (qns 73—82) 159
Definition of one-sided limits by neighbourhoods (qns 83—85) 162
Two-sided limits 163 Definition of continuity by limits (qns 86—92) 163
Theorems on limits (qns 93—99) 164
Limits as xYE and when f(x)YE (qns 100, 101) 166
Summary — continuity by neighbourhoods and limits 167
Historical note 168 Answers 171
7 Continuity and completeness: functions on intervals 182
Monotonic functions: one-sided limits (qns 1—7) 182
Intervals (qns 8—11) 183
Intermediate Value Theorem (qns 12—21) 185
Inverses of continuous functions (qns 22—28) 186
Continuous functions on a closed interval (qns 29—36) 188
Uniform continuity (qn 37—45) 190
Extension of functions on Q to functions on R (qns 46—48) 192
Summary 194
Historical note 195
Answers 197
8 Derivatives: tangents 203
Definition of derivative (qns 1—9) 203
Sums of functions (qns 10, 11) 205
The product rule(qns13—16) 205
The quotient rule(qn17) 206
The chain rule (qn18) 206
Differentiability and continuity (qns 12, 19—25) 207
Derived functions (qns 26—34) 211
Second derivatives (qns 35—38) 212
Inverse functions (qns 39—45) 213
Derivatives at end points (qn 46) 215
Summary 215
Historical note 216 Answers 219
9 Differentiation and completeness: Mean Value Theorems,Taylor’s Theorem 224
Rolle’s Theorem (qns 1—11) 224
An intermediate value theorem for derivatives (qn 12) 226
The Mean Value Theorem (qns 13—24) 226
Cauchy’s Mean Value Theorem (qn 25) 230
de l’Hoˆpital’s rule(qns26—30) 230
Summary — Rolle’s Theorem and Mean Value Theorem 232
The Second and Third Mean Value Theorems (qns 31—34) 234
Taylor’s Theorem or nth Mean Value Theorem (qns 35, 36) 235
Maclaurin’s Theorem (qns 37—46) 236
Summary — Taylor’s Theorem 239
Historical note 240
Answers 243
10 Integration: the Fundamental Theorem of Calculus 251
Areas with curved boundaries (qns 1—6) 251
Monotonic functions (qns 7—9) 254
The definite integral (qns 10—11) 255
Step functions (qns 12—15) 256
Lower integral and upper integral (qns 16—22) 258
The Riemann integral (qns 23—25) 260
Summary — definiton of the Riemann integral 261
Theorems on integrability (qns 26—36) 262
Integration and continuity (qns 37—44) 265
Mean Value Theorem for integrals (qn 45) 267
Integration on subintervals (qns 46—48) 267
Summary — properties of the Riemann integral 267
Indefinite integrals (qns 49—53) 268
The Fundamental Theorem of Calculus (qns 54—56) 270
Integration by parts (qns 57—59) 270
Integration by substitution (qn 60) 271
Improper integrals (qns 61—68) 271
Summary — the Fundamental Theorem of Calcalus 273
Historical note 273
Answers 276
11 Indices and circle functions 286
Exponential and logarithmic functions 286
Positive integers as indices (qns 1—3) 286
Positive rationals as indices(qns4—7) 287
Rational numbers as indices (qns 8 —17) 287
Real numbers as indices (qns 18—24) 289
Natural logarithms (qns 25—31) 290
Exponential and logarithmic limits (qns 32—38) 291
Summary — exponential and logarithmic functions 292
Circular or trigonometric functions 293
Length of a line segment (qns 39—42) 294
Arc length (qns 43—48) 295
Arc cosine(qns49,50) 296
Cosine and sine(qns51—58) 297
Tangent (qns 59—62) 298
Summary — circular or trigonometric functions Historical note 298
Answers 302
12 Sequences of functions 309
Pointwise limit functions (qns1—14) 310
Uniform convergence (qns 15—19) 312
Uniform convergence and continuity (qns 20—23) 313
Uniform convergence and integration (qns 24—31) 315
Summary — uniform convergence, continuity and integration 317
Uniform convergence and differentiation (qns 32—34) 318
Uniform convergence of power series (qns 35—44) 319
The Binomial Theorem for any real index (qn 45) 322
The blancmange function (qn46) 322
Summary — differentiation and the M-test 326
Historical note 326
Answers 328
Appendix 1 Properties of the real numbers 337
Appendix 2 Geometry and intuition 340
Appendix 3 Questions for student investigation and discussion 342
Bibliography 346
Index 351
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读后感
用户评价
这本书的封面设计确实很吸引人，那种深邃的蓝色调搭配简洁的字体，一下子就能让人感受到一种严谨而又深邃的学术氛围。我本来对数学分析的理解还停留在教科书上那些枯燥的定义和定理的层面，但这本书的排版和章节的组织方式，给我带来了一种全新的阅读体验。它的结构安排得非常巧妙，仿佛是在引导读者一步步深入一个迷宫，每走过一个拐角，都有一个新的视野豁然开朗。我尤其欣赏作者在引入新概念时所采用的类比手法，它们往往来源于我们日常生活中非常具体的例子，这极大地降低了初学者的理解门槛，让我感觉原本高高在上的数学理论变得触手可及。书中对一些基础概念的阐述，比如极限和连续性的引入，不再是冷冰冰的符号堆砌，而是融入了丰富的几何直觉和物理图像，这对于我这种更偏向直观理解的学习者来说，简直是及时雨。我感觉作者在写作时，不仅仅是将知识点罗列出来，更是在构建一个完整的思维体系，鼓励读者去思考“为什么”而不是仅仅记住“是什么”。这种深入浅出的教学方式，让我在阅读过程中充满了探索的乐趣，完全没有传统数学书籍常有的那种压迫感。
这本书的行文风格简直是数学写作界的一股清流。它没有陷入那种故作高深的学术腔调，而是以一种近乎于对话的口吻与读者交流。我记得读到关于级数收敛性那一部分时，作者用了很大篇幅来讨论历史背景和不同数学家为此付出的努力，这让原本抽象的证明过程一下子有了“人情味”。你会感觉到，每一个定理的诞生都不是凭空出现的，而是无数次尝试、错误和灵感碰撞的结果。这种叙事性的写作手法，极大地激发了我对数学史的好奇心，让我不再仅仅把数学视为一套死板的工具箱，而是一个充满活力的、不断发展的学科。更令人称道的是，书中对例题的选择和解析。它们不是那种一眼就能看出解法的套路题，而是精心挑选的、能够体现核心思想的“探针”。作者在解析这些例题时，会详细剖析每一步推理背后的逻辑跳跃点，并且会给出多种解题思路的比较，这对于培养我的批判性思维非常有帮助。我常常在读完一个例题的详细解答后，会合上书本，尝试用自己刚学到的工具去解决一个稍微变种的问题，这种主动学习的效率远超被动接受。
如果用一个词来概括这本书带给我的影响，那大概是“结构化重塑”。在我翻开这本书之前，我对微积分和分析的理解是零散的，知识点像散落的珍珠。这本书就像一根精心打磨的丝线，将这些珍珠串联成了一条清晰、有逻辑的项链。作者在处理诸如紧凑性、一致收敛性这些核心概念时，总是能巧妙地回溯到之前建立的基础之上，使得每一个新概念的出现都显得水到渠成，而非凭空出现。这种自洽的体系构建能力是这本书最大的价值所在。它教会了我如何像一个真正的数学家那样去思考：如何从最基本的公理出发，构建起宏伟的理论大厦，并且确保每一个支撑点都坚固可靠。我感觉自己不再是被动地接受结论，而是主动地参与到数学构建的过程中。对于那些希望彻底理解数学分析内在逻辑、并准备向更高阶数学领域迈进的读者来说，这本书提供了一个极其坚实且富有启发性的基石，它的价值远超其书本定价。
这本书在视觉呈现和排版设计上，体现出一种极高的专业水准。纸张的质量非常好，墨色深沉，即便是长时间阅读也不会感到眼睛疲劳，这对于一本动辄数百页的数学著作来说至关重要。更让我惊喜的是图表的运用。在描述像魏尔斯特拉斯处处不连续函数这类反直觉的概念时，图表的辅助作用是决定性的。作者绘制的插图清晰、比例准确，并且富有信息量，它们不仅仅是装饰，而是证明链条中不可分割的一部分。我发现，每当我对某个抽象定义感到困惑时，回头看看相应的图示，往往能立刻找到那个“啊哈！”的瞬间。这种对细节的关注延伸到了数学符号的使用上。符号的字体选择、间距的设置，都经过了精心的考量，保证了公式在视觉上具有极佳的可读性和美感，这在很大程度上提升了阅读的流畅性，避免了因符号识别困难而打断思维进程的情况。总而言之，这是一本从触感到智力都得到尊重的书籍。
我得说，这本书的严谨性是无可挑剔的，但它的“严谨”是以一种非常友好和负责任的方式呈现出来的。许多其他分析教材在处理一些关键的拓扑性质时，往往会轻描淡写地一带而过，美其名曰“留给读者自行验证”，但这对于自学者来说简直是灾难。然而，在本书中，即便是那些看似微不足道的细节，作者也给予了充分的关注。比如，在讨论反常积分和傅里叶级数之间联系的时候，涉及到了一些更深层次的函数空间概念，作者没有回避，而是用一种非常克制但又足够深入的方式进行了阐述，确保读者在建立直观理解的同时，不会在严格性上留下隐患。我特别喜欢作者在章节末尾设置的“思考与挑战”部分。它们不像传统的习题那样只是检验你是否记住了公式，而是真正需要你运用数学思维去构筑论证的“小项目”。完成其中几个挑战后，我感觉自己对整个分析框架的掌握又上了一个台阶，那种成就感是看几遍标准证明都无法替代的。它真正做到了“授人以渔”。
