Uniform Continuity

analysis

Definition

Uniform Continuity

A function $f:D\to \mathbb{R}$ is called uniformly continuous on $D$ if for every $\epsilon >0$, there exists a $\delta >0$ such that for all $x_1,x_2\in D$:

$$|x_1-x_2|<\delta ⟹|f(x_1)-f(x_2)|<\epsilon$$

Unlike ordinary continuity, the choice of $\delta$ depends only on $\epsilon$ and not on the specific point $x$ in the domain.

Properties

• Implication: Uniform continuity implies continuity, but the converse is not true (e.g., $f(x)=\frac{1}{x}$ on $(0,1]$ is continuous but not uniformly continuous).
• Heine-Cantor Theorem: If $f$ is continuous on a compact domain, then $f$ is uniformly continuous.
• Lipschitz Continuity: Every Lipschitz continuous function is uniformly continuous.
• Extension: A uniformly continuous function on a bounded interval can be uniquely extended to a continuous function on the closure of that interval.