Continuous Functions of Vector Variables

• Alberto Guzman

1. Front Matter
Pages i-x
2. Alberto Guzman
Pages 1-31
3. Alberto Guzman
Pages 33-53
4. Alberto Guzman
Pages 55-84
5. Alberto Guzman
Pages 85-118
6. Alberto Guzman
Pages 119-153
7. Back Matter
Pages 155-209

Introduction

This text is appropriate for a one-semester course in what is usually called ad­ vanced calculus of several variables. The focus is on expanding the concept of continuity; specifically, we establish theorems related to extreme and intermediate values, generalizing the important results regarding continuous functions of one real variable. We begin by considering the function f(x, y, ... ) of multiple variables as a function of the single vector variable (x, y, ... ). It turns out that most of the n treatment does not need to be limited to the finite-dimensional spaces R , so we will often place ourselves in an arbitrary vector space equipped with the right tools of measurement. We then proceed much as one does with functions on R. First we give an algebraic and metric structure to the set of vectors. We then define limits, leading to the concept of continuity and to properties of continuous functions. Finally, we enlarge upon some topological concepts that surface along the way. A thorough understanding of single-variable calculus is a fundamental require­ ment. The student should be familiar with the axioms of the real number system and be able to use them to develop elementary calculus, that is, to define continuous junction, derivative, and integral, and to prove their most important elementary properties. Familiarity with these properties is a must. To help the reader, we provide references for the needed theorems.

Keywords

Derivative Functional Analysis Linear Multivariable Calculus calculus compactness ksa

Authors and affiliations

• Alberto Guzman
• 1
1. 1.Department of MathematicsThe City College of New York, CUNYNew YorkUSA

Bibliographic information

• DOI https://doi.org/10.1007/978-1-4612-0083-3
• Copyright Information Birkhäuser Boston 2002
• Publisher Name Birkhäuser, Boston, MA
• eBook Packages
• Print ISBN 978-0-8176-4273-0
• Online ISBN 978-1-4612-0083-3
• Buy this book on publisher's site