Nonsmooth Analysis is a relatively recent area of mathematical analysis. The literature about this subject consists mainly in research papers and books. The purpose of this book is to provide a handbook for undergraduate and graduate students of mathematics that introduce this interesting area in detail.

Preface ix

Acknowledgment xv

Chapter 1. Basic Concepts and Results 1

1.1. Upper and Lower Limits 1

1.2. Semicontinuity 3

1.3. Differentiability 8

1.4. Two Important Theorems 10

1.5. Problems 14

Chapter 2. Convex Functions 17

2.1. Convex Sets and Convex Functions 17

2.2. Continuity of Convex Functions 23

2.3. Separation Results 28

2.4. Convexity and Differentiability 31

2.5. Problems 34

Chapter 3. The Subdifferential of a Convex Function 37

3.1. Subdifferential Properties 37

3.2. Two Examples 42

3.3. Problems 44

Chapter 4. The Subdifferential: General Case 47

4.1. Definition and Basic Properties 47

4.2. Geometrical Meaning of the Subdifferential 54

4.3. Density of Subdifferentiability Points 64

4.4. Proximal Subdifferential 66

4.5. Problems 68

Chapter 5. Calculus 73

5.1. Sum Rule 73

5.2. Constrained Minima 77

5.3. Chain Rule 79

5.4. Regular Functions: Elementary Properties 83

5.5. Mean Value Results 85

5.6. Decreasing Functions 88

5.7. Problems 95

Chapter 6. Lipschitz Functions and the Generalized Gradient 99

6.1. Lipschitz Regular Functions 99

6.2. The Generalized Gradient 106

6.3. Generalized Jacobian 116

6.4. Graphical Derivative 120

6.5. Problems 124

Chapter 7. Applications 129

7.1. Flow Invariant Sets 129

7.2. Viscosity Solutions 132

7.3. Solving Equations 137

7.4. Problems 143

Bibliography 145

Index 147

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