**Mathematical Analysis Vol 1 (2011)**

A top quality open textbook for advanced undergraduates and graduate students.

A complementary textbook to Saylor Academy’s Free Online Math 241 course.

Purchase Print $**28.92,** 365 pages. Softcover, black and white.

Download free pdf, 4mb, 385 pages.

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## Mathematical Analysis Vol 1, (2011)

** “Starred” sections may be omitted by beginners.*

### Chapter 1. Set Theory

1–3. Sets and Operations on Sets. Quantifiers 1

*Problems in Set Theory 6*

4–7. Relations. Mappings 8

*Problems on Relations and Mappings 14*

8. Sequences 15

9. Some Theorems on Countable Sets 18

*Problems on Countable and Uncountable Sets 21*

### Chapter 2. Real Numbers. Fields 23

1–4. Axioms and Basic Definitions 23

5–6. Natural Numbers. Induction 27

*Problems on Natural Numbers and Induction 32*

7. Integers and Rationals 34

8–9. Upper and Lower Bounds. Completeness 36

*Problems on Upper and Lower Bounds 40*

10. Some Consequences of the Completeness Axiom 43

11–12. Powers With Arbitrary Real Exponents. Irrationals 46

*Problems on Roots, Powers, and Irrationals 50*

13. The Infinities. Upper and Lower Limits of Sequences 53

*Problems on Upper and Lower Limits of Sequences in E* 60*

### Chapter 3. Vector Spaces. Metric Spaces 63

1–3. The Euclidean n-space, En 63

*Problems on Vectors in En 69*

4–6. Lines and Planes in En 71

*Problems on Lines and Planes in En 75*

7. Intervals in En 76

*Problems on Intervals in En 79*

8. Complex Numbers 80

*Problems on Complex Numbers 83*

*9. Vector Spaces. The Space Cn. Euclidean Spaces 85

*Problems on Linear Spaces 89*

*10. Normed Linear Spaces 90

*Problems on Normed Linear Spaces 93*

11. Metric Spaces 95

Problems on Metric Spaces 98

12. Open and Closed Sets. Neighborhoods 101

Problems on Neighborhoods, Open and Closed Sets 106

13. Bounded Sets. Diameters 108

Problems on Boundedness and Diameters 112

14. Cluster Points. Convergent Sequences 114

Problems on Cluster Points and Convergence 118

15. Operations on Convergent Sequences 120

Problems on Limits of Sequences 123

16. More on Cluster Points and Closed Sets. Density 135

Problems on Cluster Points, Closed Sets, and Density 139

17. Cauchy Sequences. Completeness 141

Problems on Cauchy Sequences 144

### Chapter 4. Function Limits and Continuity 149

1. Basic Definitions 149

Problems on Limits and Continuity 157

2. Some General Theorems on Limits and Continuity 161

More Problems on Limits and Continuity 166

3. Operations on Limits. Rational Functions 170

Problems on Continuity of Vector-Valued Functions 174

4. Infinite Limits. Operations in E* 177

Problems on Limits and Operations in E* 180

5. Monotone Functions 181

Problems on Monotone Functions 185

6. Compact Sets 186

Problems on Compact Sets 189

*7. More on Compactness 192

8. Continuity on Compact Sets. Uniform Continuity 194

Problems on Uniform Continuity; Continuity on Compact Sets. 200

9. The Intermediate Value Property 203

Problems on the Darboux Property and Related Topics 209

10. Arcs and Curves. Connected Sets 211

Problems on Arcs, Curves, and Connected Sets 215

*11. Product Spaces. Double and Iterated Limits 218

*Problems on Double Limits and Product Spaces 224

12. Sequences and Series of Functions 227

Problems on Sequences and Series of Functions 233

13. Absolutely Convergent Series. Power Series 237

More Problems on Series of Functions 245

### Chapter 5. Differentiation and Antidifferentiation 251

1. Derivatives of Functions of One Real Variable 251

Problems on Derived Functions in One Variable 257

2. Derivatives of Extended-Real Functions 259

Problems on Derivatives of Extended-Real Functions 265

3. L’Hˆopital’s Rule 266

Problems on L’Hˆopital’s Rule 269

4. Complex and Vector-Valued Functions on E1 271

Problems on Complex and Vector-Valued Functions on E1 275

5. Antiderivatives (Primitives, Integrals) 278

Problems on Antiderivatives 285

6. Differentials. Taylor’s Theorem and Taylor’s Series 288

Problems on Taylor’s Theorem 296

7. The Total Variation (Length) of a Function f : E1 ? E 300

Problems on Total Variation and Graph Length 306

8. Rectifiable Arcs. Absolute Continuity 308

Problems on Absolute Continuity and Rectifiable Arcs 314

9. Convergence Theorems in Differentiation and Integration 314

Problems on Convergence in Differentiation and Integration 321

10. Sufficient Condition of Integrability. Regulated Functions 322

Problems on Regulated Functions 329

11. Integral Definitions of Some Functions 331

Problems on Exponential and Trigonometric Functions 338

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