uniform continuity

一致连续

Analysis, Measure & PDE · Undergraduate · Mathematics Resources editorial compilation (AI-assisted)

Translation evidence: Literature usage

Meaning in context

A map f between metric spaces is uniformly continuous if for every epsilon > 0 there is delta > 0 such that d(x,y) < delta implies d(f(x),f(y)) < epsilon for every pair of domain points.

度量空间间的映射若对每个正误差都可选取一个适用于定义域所有点对的正距离阈值，则称为一致连续。

Example

$f(x)=3x$ on $\mathbb R$ works with $\delta=\varepsilon/3$.

Usage and distinctions

The input threshold is independent of the base point; ordinary continuity allows that dependence.

Read the encyclopedia explanation

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Michael Taylor — Introduction to Analysis in One Variable§3.1, equations (3.1.10)–(3.1.12) · Checked 2026-10-14
2. John M. Erdman — Functional Analysis and Operator Algebras: An Introduction§3.2, Definition 3.2.4 (uniform continuity between metric spaces), printed p.26 · Checked 2026-10-15

Translation sources

1. 北京大学数学科学学院 — 数学分析（实验班）课程介绍第一部分 一元微积分学，三、函数的极限与连续性：一致连续函数 · Checked 2026-10-14

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