continuous function

连续函数

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

Translation evidence: Literature usage

Meaning in context

A map between metric spaces is continuous if, at every domain point, every positive output tolerance is met by some positive input tolerance.

度量空间之间的映射，若在定义域每一点，对任意正的输出误差都能选取正的输入误差使条件成立，则称为连续函数。

Example

$f(x)=x^2$ is continuous on $\mathbb R$.

Usage and distinctions

The tolerance may depend on the point; uniform continuity removes 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, continuity definition and Proposition 3.1.1 · Checked 2026-10-14
2. John M. Erdman — Functional Analysis and Operator Algebras: An Introduction§2.1, Definition 2.1.9 (continuity by open preimages), together with §3.1, Example 3.1.34 (metric topology by open balls) · Checked 2026-10-15

Translation sources

1. 中国矿业大学数学学院 — 数学分析讲义，第四章§1 一、函数在一点的连续性; 三、区间上的连续函数 · Checked 2026-10-14

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