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# Uniform Continuity of Functions Uniform continuity is a stronger form of continuity that applies uniformly over the entire domain of a function. ## Definition A function $f: A \to \mathbb{R}$ is **uniformly continuous** on $A$ if for every $\varepsilon > 0$, there exists a $\delta > 0$ such that for all $x, y \in A$, $$|x - y| 0$, choose $\delta = \frac{\varepsilon}{2}$. Then for any $x,y$, $$|x-y|