Continuity and Uniform Continuity: Explained with Examples

A function f is continuous at a point c when small changes in the input produce small changes in the output: for every ε > 0 there is a δ > 0 such that |x − c| < δ implies |f(x) − f(c)| < ε. The function is uniformly continuous on a set when a single δ works for every pair of points at once: for every ε > 0 there is a δ > 0 such that |x − y| < δ implies |f(x) − f(y)| < ε for all x, y in the set.

The difference is subtle but crucial: ordinary continuity lets δ depend on the location c, while uniform continuity demands one δ for the whole domain. Uniform continuity is what guarantees that a function cannot oscillate wildly or blow up as you move across its domain, which is exactly what integration theory needs.

Key Concepts

• Continuity at c: limx→c f(x) = f(c). Equivalently, the ε–δ statement above with δ allowed to depend on c.
• Uniform continuity: the same ε–δ statement, but δ depends only on ε and works simultaneously at every point of the domain.
• Heine–Cantor theorem: a continuous function on a closed bounded interval [a, b] is automatically uniformly continuous there.
• Continuous but not uniformly: f(x) = 1/x on (0, 1) is continuous at every point, yet no single δ works near 0, because the function gets arbitrarily steep.
• Lipschitz functions: if |f(x) − f(y)| ≤ K|x − y| for some constant K, then f is uniformly continuous with δ = ε/K. Every such function is "uniformly tame".

Formulas

• Continuity at c: ∀ε > 0, ∃δ > 0: |x − c| < δ ⇒ |f(x) − f(c)| < ε (δ may depend on ε and c).
• Uniform continuity on D: ∀ε > 0, ∃δ > 0: ∀x, y ∈ D, |x − y| < δ ⇒ |f(x) − f(y)| < ε (δ depends only on ε).
• Lipschitz test: bounded derivative on an interval implies Lipschitz: |f(x) − f(y)| ≤ (max |f′|)|x − y|, hence uniform continuity.
• Failure test: find sequences xn, yn with |xn − yn| → 0 but |f(xn) − f(yn)| staying above some fixed ε₀; then f is not uniformly continuous.

Worked Example

Prove that f(x) = 2x + 3 is uniformly continuous on the whole real line.

1. Let ε > 0 be given. We need one δ, independent of position, such that |x − y| < δ always gives |f(x) − f(y)| < ε.
2. Compute the output difference: |f(x) − f(y)| = |(2x + 3) − (2y + 3)| = 2|x − y|.
3. We need 2|x − y| < ε, i.e. |x − y| < ε/2. Choose δ = ε/2.
4. Verification: whenever |x − y| < ε/2, we have |f(x) − f(y)| = 2|x − y| < 2·(ε/2) = ε, for every pair x, y. The same δ works everywhere, so f is uniformly continuous.

Common Misconceptions

• "Continuity on an interval automatically means uniform continuity." Only on closed bounded intervals. On (0, 1), the function 1/x is continuous but not uniformly continuous.
• "Uniform continuity is about the function being smooth." It is about the modulus of continuity being location-independent; f(x) = √x on [0, 1] is uniformly continuous despite its vertical tangent at 0.
• "Differentiable implies uniformly continuous." Differentiable implies continuous, but x² is differentiable on ℝ and not uniformly continuous there, since its slope grows without bound.

Real-World Applications

• Riemann integration: uniform continuity on [a, b] is the key lemma proving that every continuous function on a closed interval is integrable.
• Numerical analysis: uniform continuity gives explicit error control — one δ guarantees accuracy everywhere, not just near a chosen point.
• Extending functions: a uniformly continuous function on a bounded open interval extends continuously to the endpoints, which fails for merely continuous functions like sin(1/x).

Continuity lets the allowed input window δ change from point to point; uniform continuity insists on one δ for the entire domain. On closed bounded intervals the two coincide thanks to Heine–Cantor, but on open or unbounded domains they can differ sharply, as 1/x on (0, 1) shows. Whenever you need error control that does not depend on location, uniform continuity is the property to check.