1
351
Introduction to Mathematical Analysis I - 3rd Edition
TOOLS FOR ANALYSIS
183
Basic Concepts of Set Theory
183
Functions
187
The Natural Numbers and Mathematical Induction
191
Ordered Field Axioms
195
The Completeness Axiom for the Real Numbers
199
Applications of the Completeness Axiom
203
SEQUENCES
209
Convergence
209
Limit Theorems
217
Monotone Sequences
221
The Bolzano-Weierstrass Theorem
227
Limit Superior and Limit Inferior
230
LIMITS AND CONTINUITY
237
Limits of Functions
237
Limit Theorems
243
Continuity
250
Properties of Continuous Functions
254
Uniform Continuity
259
DIFFERENTIATION
265
Definition and Basic Properties of the Derivative
265
The Mean Value Theorem
271
Some Applications of the Mean Value Theorem
278
L'Hôpital's Rule
280
Taylor's Theorem
287
ADDITIONAL TOPICS
291
Topology of the Real Line
291
Continuity and Compactness
298
Limit Superior and Limit Inferior of Functions
301
Lower Semicontinuity and Upper Semicontinuity
307
Convex Functions and Derivatives
312
Nondifferentiable Convex Functions and Subdifferentials
317
References
327
Solutions and Hints for Selected Exercises
329
cover-sheet_3rd-edition.pdf
178
Accessibility Statement
178
INTRODUCTION_TO_MATHEMATICAL_ANALYSIS_I_3rd_ed.pdf
8
TOOLS FOR ANALYSIS
8
Basic Concepts of Set Theory
8
Functions
12
The Natural Numbers and Mathematical Induction
16
Ordered Field Axioms
20
The Completeness Axiom for the Real Numbers
24
Applications of the Completeness Axiom
28
SEQUENCES
34
Convergence
34
Limit Theorems
42
Monotone Sequences
46
The Bolzano-Weierstrass Theorem
52
Limit Superior and Limit Inferior
55
LIMITS AND CONTINUITY
62
Limits of Functions
62
Limit Theorems
68
Continuity
75
Properties of Continuous Functions
79
Uniform Continuity
84
DIFFERENTIATION
90
Definition and Basic Properties of the Derivative
90
The Mean Value Theorem
96
Some Applications of the Mean Value Theorem
103
L'Hôpital's Rule
105
Taylor's Theorem
112
ADDITIONAL TOPICS
116
Topology of the Real Line
116
Continuity and Compactness
123
Limit Superior and Limit Inferior of Functions
126
Lower Semicontinuity and Upper Semicontinuity
132
Convex Functions and Derivatives
137
Nondifferentiable Convex Functions and Subdifferentials
142
References
152
Solutions and Hints for Selected Exercises
154
cover-sheet_3rd-edition.pdf
3
Accessibility Statement
3
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