Sets, Sequences and Mappings

The Basic Concepts of Analysis

Kenneth Anderson, Dick Wick Hall

• Publisher: Dover Publications
• Year: 2012
• ISBN:
  9780486158129
• Topic: Mathematics
• Subtopic: Calculus
• Index: Mathematics

Students progressing to advanced calculus are frequently confounded by the dramatic shift from mechanical to theoretical and from concrete to abstract. This text bridges the gap, offering a systematic development of the real number system and careful treatment of mappings, sequences, limits, continuity, and metric spaces.
The first five chapters consist of a systematic development of many of the important properties of the real number system, plus detailed treatment of such concepts as mappings, sequences, limits, and continuity. The sixth and final chapter discusses metric spaces and generalizes many of the earlier concepts and results involving arbitrary metric spaces.
An index of axioms and key theorems appears at the end of the book, and more than 300 problems amplify and supplement the material within the text. Geared toward students who have taken several semesters of basic calculus, this volume is an ideal prerequisite for mathematics majors preparing for a two-semester course in advanced calculus.

V

Continuity and uniform continuity

1. Continuous functions

We have already defined and discussed the notion of a mapping ƒ from a set A into (or onto) a set B, where A and B are sets in a given universe. Now, in the special case (as in most of our work thus far) where the universe is R₁, we see that both A and B are subsets of R₁; that is, both the domain and the range of ƒ are sets of real numbers. Since the image of x under ƒ is a real number, we often speak of ƒ as a real valued function, or more generally, a real-valued mapping. We often shorten this by referring to ƒ as a function, although the terms “mapping” and “function” are used interchangeably.

We stated in Chapter III that one of the important objectives in analysis is the determination of those properties of sets which are preserved under certain types of mappings. As a case in point, we showed that onto mappings preserve countability. Since Chapter IV is devoted to a detailed study of convergent sequences, we might well wonder what kind of mapping preserves convergent sequences. We call such a mapping a continuous mapping and define it as follows.

Definition 1.1. Let A ⊂ R₁, let ƒ be a function ƒ: A → R₁, and let p be any point in A. We say that ƒ is continuous at the point p iff given any sequence of points 〈p_n〉 of A which converges to the point p, then the sequence of images 〈ƒ(p_n)〉converges to the point ƒ(p). We say that ƒ is continuous on the set A iff ƒ is continuous at the point p for every p in A. The statement “ƒ is continuous,” where ƒ: A → R₁ means “ƒ is continuous on A.”

As an immediate consequence of Definition 1.1, we have the following two theorems.

Theorem 1.2. Every constant function is continuous on R₁.

Proof. Let ƒ: R₁ → R₁ be defined by ƒ(x) = k for every x ε R₁ where k is a real number. We want to show that ƒ is continuous on R₁. Thus, let p be any point in R₁, and let 〈p_n〉 be any sequence of points in R₁ such that lim p_n = p. By definition of ƒ, we see that ƒ(p_n) = k for every n, and also, ƒ(p) = k. Thus, we have lim ƒ(p_n) = k = ƒ(p). Therefore ƒ...

Table of contents

• DOVER BOOKS ON MATHEMATICS
• Title Page
• Copyright Page
• Preface
• Table of Contents
• I - Introduction to sets and mappings
• II - Sequences
• III - Countable, connected, open, and closed sets
• IV - Convergence
• V - Continuity and uniform continuity
• VI - Metric spaces
• Index of Axioms and Key Theorems
• Index
• A CATALOG OF SELECTED DOVER BOOKS IN SCIENCE AND MATHEMATICS