this text is an axiomatic treatment of the properties of continuous multivariable functions and related results from topology. in the context of normed vector spaces, the author covers boundedness, extreme values, and uniform continuity of functions, along with the connections between continuity and topological concepts such as connectedness and compactness. the order of topics deliberately mimics the order of development in elementary calculus. this sequencing allows for an elementary approach, with analogies to and generalizations from such familiar ideas as the pythagorean theorem. the reader is frequently reminded that the pictures suggested by geometry are powerful guides and tools. the definition-theorem-proof format resides within an informal exposition, containing numerous historical comments and questions within and between the proofs. the objective is to present precise proofs, but in a structure and tone that teach the student to plan and write proofs, both in general and specifically for the real analysis course that will follow this one. applications are included where they provide interesting illustrations of the principles and theorems presented. problems, solutions, bibliography and index complete this book. `continuous functions of vector variables is suitable for a course in multivariable calculus aimed at advanced undergraduates preparing for graduate programs in pure mathematics. required background includes a course in the theory of single-variable calculus and the elements of linear algebra. also by the author: derivatives and integrals of multivariable functions, isbn 0-8176-4274-9.