Continuous Functions of Vector Variables

This is an axiomatic treatment of the properties of continuous multivariable functions and related results from topology. The author covers boundedness, extreme values, and uniform continuity of functions, along with connections between continuity and topological concepts such as connectedness and compactness. The order of topics mimics the order of development in elementary calculus, with analogies and generalizations from such familiar ideas as the Pythagorean theorem.

"The book is devoted to mathematical analysis in multi-dimensional spaces, including infinite-dimensional ones…One of the important characteristics of the text is the topological approach. …It is worth mentioning that all these ideas are presented in conformity with the situation, i.e. they are strictly related to normed spaces and no unnecessary abstract setting is used. … The language is very clear and sometimes very ‘fresh’ if compared with classical textbooks, so that it may better appeal to the reader. There are many exercises, and the book concludes with a chapter containing solutions." —MATHEMATICAL REVIEWS "This is a textbook on calculus of several variables. It covers algebraic and metric structure of the Euclidean space, convergence, basic properties of continuous functions and topology in normed spaces.... In my opinion, this topological approach is one of the advantages of the book under review. It is written very clearly and contains numerous examples and instructive pictures. Each section is endowed with a set of exercises, and the book is concluded with solutions to these exercises. This is a nice and user-friendly textbook which can be recommended for a one-semester course in multivariable calculus." —ZENTRALBLATT MATH "The presentation is detailed and clear. The leisurely discursive style adopted by the author will be appreciated by beginners. The fact that the solution of each and every exercise can be found at the end of the book, considerable enhances its value and makes it suitable also for individual study. This book successfully bridges the gap between elemntary calculus and such higher disciplines as real functions, general topology, and, in particular, functional analysis." —PUBLICATIONES MATHEMATICAE, DEBRECEN