Continuous On A Compact Set Is Uniformly Continuous

Continuous on a Compact Set is Uniformly Continuous: A Comprehensive Guide

The theorem stating that a function continuous on a compact set is uniformly continuous is a cornerstone of real analysis. It elegantly connects the concepts of continuity, compactness, and uniform continuity, providing powerful tools for understanding the behavior of functions. This article will delve deep into this theorem, exploring its proof, implications, and applications, aiming for a comprehensive understanding accessible to both beginners and those seeking a more rigorous treatment.

Understanding the Key Concepts

Before diving into the proof, let's clarify the fundamental concepts involved:

1. Continuity

A function f: A → ℝ is continuous at a point x₀ ∈ A if for every ε > 0, there exists a δ > 0 such that if |x - x₀| < δ and x ∈ A, then |f(x) - f(x₀)| < ε. Intuitively, this means that the function's values stay arbitrarily close to f(x₀) as x approaches x₀. A function is continuous on a set A if it's continuous at every point in A.

2. Uniform Continuity

A function f: A → ℝ is uniformly continuous if for every ε > 0, there exists a δ > 0 such that for all x, y ∈ A, if |x - y| < δ, then |f(x) - f(y)| < ε. The crucial difference between continuity and uniform continuity lies in the quantifiers. In continuity, δ depends on both ε and the point x₀. In uniform continuity, δ depends only on ε, making the condition far stronger. It guarantees that the function's values remain close together for points sufficiently close to each other, regardless of their location in the set A.

3. Compactness

A set A ⊂ ℝ is compact if every open cover of A has a finite subcover. This seemingly abstract definition has powerful consequences. In ℝ, a set is compact if and only if it is closed and bounded. This equivalence simplifies the understanding significantly for many applications. A closed set contains all its limit points, while a bounded set is contained within some interval [-M, M] for some M > 0.

The Theorem: Continuous on a Compact Set Implies Uniform Continuity

Theorem: If a function f: A → ℝ is continuous on a compact set A, then f is uniformly continuous on A.

This theorem is not intuitive at first glance. Continuity at each point doesn't inherently guarantee uniform continuity across the entire set. Compactness provides the crucial extra condition that bridges this gap.

Proof of the Theorem

We'll present a proof by contradiction, leveraging the properties of compact sets.

1. Assume the negation: Suppose f is continuous on a compact set A, but f is not uniformly continuous.

2. Formalize the negation of uniform continuity: If f is not uniformly continuous, then there exists an ε > 0 such that for every δ > 0, there exist x, y ∈ A with |x - y| < δ but |f(x) - f(y)| ≥ ε.

3. Construct sequences: Let's choose a sequence of δ values, say δₙ = 1/n for n = 1, 2, 3, ... . For each δₙ, there exist points xₙ and yₙ in A such that |xₙ - yₙ| < 1/n and |f(xₙ) - f(yₙ)| ≥ ε.

4. Bolzano-Weierstrass Theorem: Since A is compact (and therefore bounded and closed), the sequence {xₙ} has a convergent subsequence {xₙₖ} that converges to a limit x ∈ A (because A is closed).

5. Convergence of the corresponding subsequence: Because |xₙₖ - yₙₖ| < 1/nₖ and nₖ → ∞ as k → ∞, we have that the subsequence {yₙₖ} also converges to x.

6. Continuity at the limit point: Since f is continuous at x, for our chosen ε > 0, there exists a δ > 0 such that if |z - x| < δ, then |f(z) - f(x)| < ε/2.

7. Contradiction: Since both {xₙₖ} and {yₙₖ} converge to x, there exists a K such that for all k > K, |xₙₖ - x| < δ/2 and |yₙₖ - x| < δ/2. Therefore, |xₙₖ - yₙₖ| < δ. But by the triangle inequality, |f(xₙₖ) - f(yₙₖ)| ≤ |f(xₙₖ) - f(x)| + |f(x) - f(yₙₖ)| < ε/2 + ε/2 = ε. This contradicts our assumption that |f(xₙ) - f(yₙ)| ≥ ε for all n.

8. Conclusion: Our assumption that f is not uniformly continuous must be false. Therefore, f is uniformly continuous on A.

Implications and Applications

The theorem's implications are far-reaching:

• Simplifying proofs: Many proofs become significantly easier when dealing with uniformly continuous functions on compact sets. The uniform nature of the continuity allows for broader estimations and simplifications.

• Extending results: Results proved for uniformly continuous functions can often be extended to continuous functions on compact sets.

• Numerical analysis: The theorem is crucial in numerical analysis, guaranteeing the existence of uniform approximations for continuous functions defined on compact intervals. This is fundamental in numerical integration and the solution of differential equations.

• Real analysis: The theorem highlights the deep connection between compactness, continuity, and uniform continuity within real analysis. It frequently appears in advanced analysis proofs.

Examples and Counterexamples

Example 1: Consider the function f(x) = x² on the interval [0, 1]. This interval is compact, and f(x) is continuous on [0, 1]. Therefore, by the theorem, f(x) is uniformly continuous on [0, 1].

Example 2: Consider f(x) = 1/x on the interval (0, 1). This interval is not compact (it's not closed). While f(x) is continuous on (0, 1), it is not uniformly continuous. This demonstrates the necessity of compactness in the theorem.

Example 3: Consider f(x) = sin(x) on the interval (-∞, ∞). This interval is not compact. Although f(x) is continuous, it is not uniformly continuous on this unbounded interval.

Counterexample illustrating the need for compactness: Consider the function f(x) = 1/x on the interval (0,1). This function is continuous on (0,1), but it is not uniformly continuous. The interval (0,1) is not compact because it is not closed. This counterexample clearly demonstrates that the compactness condition is essential for the theorem to hold true.

Extensions and Generalizations

The theorem can be generalized to metric spaces: If f: A → Y is a continuous function from a compact metric space A to a metric space Y, then f is uniformly continuous. This generalization extends the power and applicability of the theorem beyond the real numbers.

Conclusion

The theorem that a continuous function on a compact set is uniformly continuous is a powerful result with profound implications across various fields of mathematics. Understanding its proof and appreciating its consequences are crucial for anyone studying real analysis or related areas. The connection between compactness, continuity, and uniform continuity is a beautiful illustration of the interconnectedness of mathematical concepts and their applications in problem-solving and theoretical development. Its significance extends far beyond a mere theorem; it provides a fundamental toolset for advanced mathematical investigations and practical applications. The theorem's elegance and utility make it a cornerstone of mathematical analysis, providing a valuable lens for understanding the behavior of functions in various contexts.