The book "Convex Functions, Monotone Operators and Differentiability" by Robert R. Phelps, second edition, is a comprehensive treatment of the differentiability of convex functions and related topics in functional analysis. It covers the theory of convex functions, monotone operators, and their connections to differentiability in infinite-dimensional spaces. The book includes a detailed discussion of subdifferentials, which are set-valued maps associated with convex functions, and their relationship to monotone operators. It also explores the concept of Asplund spaces, which are Banach spaces where every continuous convex function is Gâteaux differentiable on a dense set of points. The book begins with an introduction to convex functions and their differentiability properties, including Mazur's theorem, which states that a continuous convex function on a separable Banach space is Gâteaux differentiable on a dense $ G_{\delta} $ set. The text then delves into monotone operators, their properties, and their connection to subdifferentials. It discusses the concept of Asplund spaces and their relationship to the Radon-Nikodym property (RNP), which is a key property in the study of vector measures. The book also covers variational principles, including the Borwein-Preiss smooth variational principle, and the connection between perturbed optimization and differentiability. It discusses the duality between Asplund spaces and spaces with the RNP, and presents results on the differentiability of convex functions in these spaces. The text includes a detailed exploration of lower semicontinuous convex functions, their subdifferentials, and their applications in optimization. The book is structured into seven chapters, each covering a specific topic in the field. It includes a detailed bibliography and index, making it a valuable resource for researchers and students in functional analysis and related areas. The second edition includes updates and corrections based on new developments in the field, as well as contributions from various mathematicians. The book is written in a clear and accessible style, making it suitable for graduate students and researchers interested in the theory of convex functions and monotone operators.The book "Convex Functions, Monotone Operators and Differentiability" by Robert R. Phelps, second edition, is a comprehensive treatment of the differentiability of convex functions and related topics in functional analysis. It covers the theory of convex functions, monotone operators, and their connections to differentiability in infinite-dimensional spaces. The book includes a detailed discussion of subdifferentials, which are set-valued maps associated with convex functions, and their relationship to monotone operators. It also explores the concept of Asplund spaces, which are Banach spaces where every continuous convex function is Gâteaux differentiable on a dense set of points. The book begins with an introduction to convex functions and their differentiability properties, including Mazur's theorem, which states that a continuous convex function on a separable Banach space is Gâteaux differentiable on a dense $ G_{\delta} $ set. The text then delves into monotone operators, their properties, and their connection to subdifferentials. It discusses the concept of Asplund spaces and their relationship to the Radon-Nikodym property (RNP), which is a key property in the study of vector measures. The book also covers variational principles, including the Borwein-Preiss smooth variational principle, and the connection between perturbed optimization and differentiability. It discusses the duality between Asplund spaces and spaces with the RNP, and presents results on the differentiability of convex functions in these spaces. The text includes a detailed exploration of lower semicontinuous convex functions, their subdifferentials, and their applications in optimization. The book is structured into seven chapters, each covering a specific topic in the field. It includes a detailed bibliography and index, making it a valuable resource for researchers and students in functional analysis and related areas. The second edition includes updates and corrections based on new developments in the field, as well as contributions from various mathematicians. The book is written in a clear and accessible style, making it suitable for graduate students and researchers interested in the theory of convex functions and monotone operators.