This is the sequel to Problems in Mathematical Analysis I: Real Numbers, Sequences and Series (Volume 4 in the AMS series, the Student Mathematical Library). As in the first volume, this book is divided into two parts. The first is a collection of exercises and problems, and the second contains their solutions. The book mainly deals with real functions of one real variable. Topics include: properties of continuous functions, intermediate value property, uniform continuity, mean value theorems, Taylor's formula, convex functions, sequences and series of functions. The book is mainly geared toward students studying the basic principles of analysis. However, given its selection of problems, organization, and level, it would be an ideal choice for tutorial or problem-solving seminars, particularly those geared toward the Putnam exam. It is also suitable for self-study. The presentation of material is designed to help student comprehension, to encourage them to ask their own questions, and to start research.The collection of problems in the book will also help teachers who wish to incorporate problems into their lectures. Solutions for most problems are provided.