---
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. A continuous function on a closed bounded interval is uniformly continuous"
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: 2026-09-12
---

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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. A continuous function on a closed bounded interval is uniformly continuous and 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

## Lesson: One delta for every point

A function is continuous at a point when its value exists, its limit exists, and the two are equal. Equivalently, whenever a sequence converging to the point is fed in, the outputs converge to the value there. For f(x) equal to 3x at 2, a tolerance epsilon is answered by delta equal to epsilon over 3. Most proofs chase an arbitrary tolerance back to such a distance.

Uniform continuity demands one distance delta that works at every point at once, not a fresh delta per point. The classic failure is 1 over x on (0, 1): nearer zero the graph steepens without bound, so closer points need ever smaller deltas and no single delta suits all. Tracking which quantities depend on the point is exactly what separates the two notions.

The extreme value theorem promises that a continuous function on a closed bounded interval attains a highest and a lowest value. Continuity keeps values controlled while the closed interval keeps limit points inside. Drop closedness and it breaks: f(x) equal to x on the open interval (0, 1) attains neither maximum nor minimum.

**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

- [Continuity and the Intermediate Value Theorem](https://lightmysky.com/learn/mathematics/continuity-and-the-intermediate-value-theorem-mt_kaQR6cWjbV)
- [Cauchy Sequences and the Bolzano-Weierstrass Theorem](https://lightmysky.com/learn/mathematics/cauchy-sequences-and-the-bolzano-weierstrass-theorem-mt_nreDylVkSU)

## 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)