7.1 Definition of Uniform Continuity

Uniform continuity is a key concept in mathematical analysis, strengthening the idea of continuity. It ensures that a function behaves consistently across its entire domain, maintaining a uniform level of "smoothness" or regularity.

This property is crucial for many advanced mathematical results. It bridges the gap between pointwise continuity and stronger forms of continuity, providing a powerful tool for analyzing functions in various contexts, from real analysis to topology.

Uniform continuity for functions

Definition and properties

• Uniform continuity is a property of functions between two metric spaces (X,d_X) and (Y,d_Y)
  • A function f: (X,d_X) → (Y,d_Y) is uniformly continuous if for every ε > 0, there exists a δ > 0 such that for all x, y ∈ X with d_X(x,y) < δ, we have d_Y(f(x),f(y)) < ε
  • The choice of δ depends only on ε and is independent of the points x and y in the domain
• Uniform continuity is a stronger condition than pointwise continuity
  • It requires the same δ to work for all points in the domain simultaneously
  • Every uniformly continuous function is pointwise continuous, but the converse is not true

Examples and special cases

• Lipschitz continuous functions are uniformly continuous
  • A function f is Lipschitz continuous if there exists a constant K > 0 such that |f(x) - f(y)| ≤ K|x - y| for all x, y in the domain
  • Examples of Lipschitz continuous functions include f(x) = x, f(x) = sin(x), and f(x) = x^2 on a bounded interval
• Continuous functions on compact metric spaces are uniformly continuous
  • If f: (X,d_X) → (Y,d_Y) is continuous and X is compact, then f is uniformly continuous
  • This result is known as the Heine-Cantor theorem and is a consequence of the compactness of the domain

Epsilon-delta definition of uniform continuity

Interpretation and geometric meaning

• The ε-δ definition captures the idea that the output of the function can be controlled by restricting the input
  • Given any desired level of output proximity ε > 0, we can find an input proximity δ > 0 such that whenever the distance between two inputs is less than δ, the distance between their outputs will be less than ε
  • The key aspect is that the choice of δ depends only on ε and not on the specific points in the domain, making the continuity "uniform" across the entire domain
• Geometrically, the ε-δ definition has the following interpretation
  • For any horizontal strip of width ε in the codomain, there exists a corresponding horizontal strip of width δ in the domain such that the function maps the thinner strip entirely inside the wider strip
  • This means that the function does not "oscillate" too much, as it maps nearby inputs to nearby outputs uniformly across the domain

Comparison with pointwise continuity

• Pointwise continuity is a local property, while uniform continuity is a global property
  • Pointwise continuity requires that for each point x in the domain and any ε > 0, there exists a δ > 0 (which may depend on x) such that |x - y| < δ implies |f(x) - f(y)| < ε
  • Uniform continuity requires the same δ to work for all points in the domain simultaneously, independent of the specific point
• The ε-δ definition of uniform continuity is more stringent than that of pointwise continuity
  • In pointwise continuity, the choice of δ may depend on the specific point x, while in uniform continuity, δ must work for all points in the domain
  • Consequently, every uniformly continuous function is pointwise continuous, but not every pointwise continuous function is uniformly continuous

Uniform vs pointwise continuity

Distinguishing features

• Uniform continuity is a global property, while pointwise continuity is a local property
  • Uniform continuity requires the same δ to work for all points in the domain simultaneously, independent of the specific point
  • Pointwise continuity allows the choice of δ to depend on the specific point x in the domain
• Every uniformly continuous function is pointwise continuous, but the converse is not true
  • There exist functions that are pointwise continuous but not uniformly continuous
  • An example is f(x) = 1/x on the interval (0,1], which is pointwise continuous but not uniformly continuous, as the choice of δ becomes arbitrarily small near 0

Examples and counterexamples

• The function f(x) = x^2 on the real line is both pointwise and uniformly continuous
  • For any ε > 0, we can choose δ = min{1, √ε} to satisfy the definition of uniform continuity
  • This choice of δ works for all points in the domain simultaneously
• The function f(x) = sin(x) on the real line is both pointwise and uniformly continuous
  • For any ε > 0, we can choose δ = ε to satisfy the definition of uniform continuity
  • This is because |sin(x) - sin(y)| ≤ |x - y| for all x, y ∈ ℝ (Lipschitz continuity)
• The function f(x) = 1/x on the interval (0,1] is pointwise continuous but not uniformly continuous
  • For any x ∈ (0,1] and ε > 0, we can find a δ > 0 (depending on x) such that |x - y| < δ implies |f(x) - f(y)| < ε
  • However, no single δ works for all points in the domain simultaneously, as the function becomes arbitrarily steep near 0

Proving uniform continuity

Proof strategy and techniques

• To prove a function f: (X,d_X) → (Y,d_Y) is uniformly continuous, start by considering an arbitrary ε > 0
  • The goal is to find a δ > 0 (possibly depending on ε) such that for all x, y ∈ X with d_X(x,y) < δ, we have d_Y(f(x),f(y)) < ε
  • The choice of δ should be independent of the points x and y, and should work for all pairs of points in the domain X
• Common techniques for finding δ include
  • Using the triangle inequality to break down the distance d_Y(f(x),f(y)) into manageable parts
  • Applying the mean value theorem for real-valued functions to estimate the difference |f(x) - f(y)|
  • Exploiting properties specific to the function, such as Lipschitz continuity or boundedness
• Once a suitable δ is found, the proof is complete, as the definition of uniform continuity is satisfied for the arbitrary choice of ε

Examples of proofs

• Proving that f(x) = x^2 is uniformly continuous on the real line
  • Let ε > 0 be given. We need to find a δ > 0 such that for all x, y ∈ ℝ with |x - y| < δ, we have |f(x) - f(y)| < ε
  • Using the factorization |x^2 - y^2| = |x - y| · |x + y|, we can estimate |f(x) - f(y)| ≤ |x - y| · (|x| + |y|)
  • If we restrict |x - y| < 1, then |x| ≤ |y| + 1, so |f(x) - f(y)| ≤ |x - y| · (2|y| + 1)
  • Choosing δ = min{1, ε / (2|y| + 1)} satisfies the definition of uniform continuity
• Proving that f(x) = sin(x) is uniformly continuous on the real line
  • Let ε > 0 be given. We need to find a δ > 0 such that for all x, y ∈ ℝ with |x - y| < δ, we have |f(x) - f(y)| < ε
  • Using the Lipschitz continuity of sin(x), we have |sin(x) - sin(y)| ≤ |x - y| for all x, y ∈ ℝ
  • Choosing δ = ε satisfies the definition of uniform continuity, as |x - y| < δ = ε implies |f(x) - f(y)| ≤ |x - y| < ε