---
title: "Continuity and Uniform Continuity"
description: "Continuity as a sequence or epsilon-delta condition, then the stronger version where one delta works across the whole domain. Heine-Cantor says a continuous function on a closed bounded interval is un"
canonical: https://lightmysky.com/learn/mathematics/continuity-and-uniform-continuity-mt_plWc9raOwz
source: https://lightmysky.com/learn/mathematics/continuity-and-uniform-continuity-mt_plWc9raOwz.md
retrieved: 2026-11-12
---

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# Continuity and Uniform Continuity

Continuity as a sequence or epsilon-delta condition, then the stronger version where one delta works across the whole domain. Heine-Cantor says a continuous function on a closed bounded interval is uniformly continuous, and the extreme value theorem says it attains its bounds.

Subject: Mathematics · Area: Calculus & Analysis · Ages 20 to 21
Page: https://lightmysky.com/learn/mathematics/continuity-and-uniform-continuity-mt_plWc9raOwz

## Ready when they can

- Prove continuity of a function at a point from the definition
- Give a continuous function that is not uniformly continuous, with the reason
- State the extreme value theorem and where its hypotheses are used
- Use closed-interval compactness to explain why continuity implies uniform continuity.

## Lesson: One delta for every point

A function is continuous at a point c when its value exists, its limit exists, and the two are equal. The epsilon-delta form says that for every positive epsilon there is a positive delta such that |x minus c| below delta forces |f(x) minus f(c)| below epsilon. Equivalently, every sequence converging to c produces outputs converging to f(c). For f(x) equal to 3x at 2, delta equal to epsilon over 3 works. Most proofs chase an arbitrary tolerance back to such a distance. A sequence scene can show the same idea: x_k moves toward c while f(x_k) moves toward f(c).

Uniform continuity uses one delta for every domain pair: if |x minus y| is below delta, then |f(x) minus f(y)| is below epsilon. The function e to the x on real numbers is continuous, but not uniformly continuous: far right, small input gaps create large output gaps, so no shared delta works. Pointwise continuity may choose a fresh delta at each point.

The extreme value theorem says a continuous function on a closed bounded interval [a, b] attains highest and lowest values. Points x_max and x_min in [a, b] attain them. The same hypotheses invoke Heine-Cantor: a continuous function there is uniformly continuous, so one delta works for every pair. Continuity prevents wild jumps, while closedness keeps endpoint and limit points inside. Drop closedness and it breaks: f(x) equal to x on (0, 1) attains neither bound.

**Example.** A bounded-slope example separates the ideas. Suppose a function g on all real numbers satisfies |g(x) minus g(y)| at most 2 times |x minus y|. For every epsilon, delta equal to epsilon over 2 works everywhere, so g is uniformly continuous.

**Example.** A continuous function can fail uniform continuity on an unbounded domain. For f(x) equal to x cubed on all real numbers, the output gap factors as (x minus y)(x squared plus xy plus y squared), and the second factor grows without bound. No single positive delta controls every pair.

**Recap.** Continuity matches limits to values point by point, uniformity shares one delta everywhere, and closed intervals let continuous functions attain their bounds.

## Practice

14 questions on this page, each with its working shown.

## Needs first

- [Cauchy Sequences and the Bolzano-Weierstrass Theorem](https://lightmysky.com/learn/mathematics/cauchy-sequences-and-the-bolzano-weierstrass-theorem-mt_nreDylVkSU)

## Optional connections

- [Continuity and the Intermediate Value Theorem](https://lightmysky.com/learn/mathematics/continuity-and-the-intermediate-value-theorem-mt_kaQR6cWjbV)

## Opens up

- [Baire Category and the Uniform Boundedness Principle](https://lightmysky.com/learn/mathematics/baire-category-and-the-uniform-boundedness-principle-mt_11Rd73trAX)
- [Differentiability and the Theorems Behind the Rules](https://lightmysky.com/learn/mathematics/differentiability-and-the-theorems-behind-the-rules-mt_1xLEeoA69z)
- [Metric Spaces: Distance as an Axiom](https://lightmysky.com/learn/mathematics/metric-spaces-distance-as-an-axiom-mt_A3TMb8wL6k)
- [Topological Spaces: Continuity Without Distance](https://lightmysky.com/learn/mathematics/topological-spaces-continuity-without-distance-mt_D2oPq0AVhI)
- [Counterexamples and Testing a Conjecture](https://lightmysky.com/learn/mathematics/counterexamples-and-testing-a-conjecture-mt_SjA4jqNkaM)