An Introduction to Nonsmooth Analysis - 1st Edition - ISBN: 9780128007310, 9780128008256

An Introduction to Nonsmooth Analysis

1st Edition

Authors: Juan Ferrera
eBook ISBN: 9780128008256
Paperback ISBN: 9780128007310
Imprint: Academic Press
Published Date: 26th November 2013
Page Count: 164
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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.

Key Features

  • Includes different kinds of sub and super differentials as well as generalized gradients
  • Includes also the main tools of the theory, as Sum and Chain Rules or Mean Value theorems
  • Content is introduced in an elementary way, developing many examples, allowing the reader to understand a theory which is scattered in many papers and research books


This book is mainly directed to graduate students in mathematics. It may be used as handbook for a graduate course or a reference book in an undergraduate course of advanced analysis with the aim of introduce the nonsmooth analysis as a complement to differential calculus, showing how smooth tools can be employed in the lack of differentiability.

Table of Contents




Chapter 1. Basic Concepts and Results


1.1 Upper and Lower Limits

1.2 Semicontinuity

1.3 Differentiability

1.4 Two Important Theorems

1.5 Problems

Chapter 2. Convex Functions


2.1 Convex Sets and Convex Functions

2.2 Continuity of Convex Functions

2.3 Separation Results

2.4 Convexity and Differentiability

2.5 Problems

Chapter 3. The Subdifferential of a Convex Function


3.1 Subdifferential Properties

3.2 Two Examples

3.3 Problems

Chapter 4. The Subdifferential: General Case


4.1 Definition and Basic Properties

4.2 Geometrical Meaning of the Subdifferential

4.3 Density of Subdifferentiability Points

4.4 Proximal Subdifferential

4.5 Problems

Chapter 5. Calculus


5.1 Sum Rule

5.2 Constrained Minima

5.3 Chain Rule

5.4 Regular Functions: Elementary Properties

5.5 Mean Value Results

5.6 Decreasing Functions

5.7 Problems

Chapter 6. Lipschitz Functions and the Generalized Gradient


6.1 Lipschitz Regular Functions

6.2 The Generalized Gradient

6.3 Generalized Jacobian

6.4 Graphical Derivative

6.5 Problems

Chapter 7. Applications


7.1 Flow Invariant Sets

7.2 Viscosity Solutions

7.3 Solving Equations

7.4 Problems




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© Academic Press 2014
Academic Press
eBook ISBN:
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About the Author

Juan Ferrera

Affiliations and Expertise

Universidad Complutense de Madrid, Spain


"...starting from the very beginning, adopting a slow, easy to follow linear development and reaching to a self-contained theory...oriented towards undergraduate students, as a  first quick introduction to the topic."--MathSciNet, An Introduction to Nonsmooth Analysis

"...devoted to presenting the theory of the subdifferential of lower semicontinuous functions which is a generalization of the subdifferential of convex functions...a good reference for researchers in optimization and applied mathematics."--Zentralblatt MATH, Sep-14

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