Book contents
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 The Real and Complex Numbers
- 3 Real and Complex Sequences
- 4 Series
- 5 Power Series
- 6 Metric Spaces
- 7 Continuous Functions
- 8 Calculus
- 9 Some Special Functions
- 10 Lebesgue Measure on the Line
- 11 Lebesgue Integration on the Line
- 12 Function Spaces
- 13 Fourier Series
- 14 * Applications of Fourier Series
- 15 Ordinary Differential Equations
- Appendix: The Banach-Tarski Paradox
- Hints for Some Exercises
- Notation Index
- General Index
7 - Continuous Functions
Published online by Cambridge University Press: 06 July 2010
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 The Real and Complex Numbers
- 3 Real and Complex Sequences
- 4 Series
- 5 Power Series
- 6 Metric Spaces
- 7 Continuous Functions
- 8 Calculus
- 9 Some Special Functions
- 10 Lebesgue Measure on the Line
- 11 Lebesgue Integration on the Line
- 12 Function Spaces
- 13 Fourier Series
- 14 * Applications of Fourier Series
- 15 Ordinary Differential Equations
- Appendix: The Banach-Tarski Paradox
- Hints for Some Exercises
- Notation Index
- General Index
Summary
The notion of continuity of a function at a point makes sense whenever both the domain and the range of the function are metric spaces. Continuity and related issues are discussed from a general point of view in this chapter. In some circumstances, continuous functions themselves constitute a natural metric space. The Weierstrass Polynomial Approximation Theorem can be viewed in this context.
Definitions and General Properties
Suppose that S and T are two sets. By a function f from S to T we mean an assignment to each point p ∈ S of a unique point of T, denoted f(p). The associated notation is f : S → T, which is read “f is a function from S to T,” or “f maps S to T.”
Suppose that S and T are metric spaces and f is a function from S to T. The function f is said to be continuous at p if for each ε > 0 there is δ > 0 such that dS(p, p′) < δ implies dT (f(p), f(p′)) < ε. The function f is said to be continuous if it is continuous at each point of S. If A is a subset of S and g is a function from A to T, then we extend these definitions to g by considering A itself as a metric space, with the metric it inherits from S.
From now on we take S and T to be metric spaces and derive a number of results that are exercises in the use of the various definitions.
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- AnalysisAn Introduction, pp. 86 - 98Publisher: Cambridge University PressPrint publication year: 2004