An Introduction to Nonsmooth Analysis book cover

An Introduction to Nonsmooth Analysis

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.


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.

Paperback, 164 Pages

Published: November 2013

Imprint: Academic Press

ISBN: 978-0-12-800731-0


  • "...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


  • Chapter 1. Basic concepts and results: Upper and lower limits. Semicontinuity. Differentiability. Two important Theorems.
    Chapter 2. Convex Functions: Convex sets and convex functions. Continuity of convex functions. Separation Results. Convexity and Differentiability.
    Chapter 3. The subdifferential of a Convex function: Subdifferential properties.  Examples.
    Chapter 4. The subdifferential. General case: Definition and basic properties. Geometrical meaning of the subdifferential. Density of subdifferentiability points. Proximal subdifferential
    Chapter 5. Calculus: Sum Rule. Constrained minima. Chain Rule. Regular functions: Elementary properties. Mean Value results. Decreasing Functions
    Chapter 6. Lipschitz functions and the generalized gradient: Lipschitz regular functions. The generalized gradient. Generalized Jacobian. Graphical derivative
    Chapter 7. Applications: Flow invariant sets. Viscosity solutions. Solving equations.


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