# Definition(s) > [!NOTE] Definition (Uniformly continuous function) > Let $A \subset \mathbb{R}.$ A [[Real Function|real function]] $f:A\to \mathbb{R}$ is *uniformly continuous* iff for all $\varepsilon>0,$ there exists $\delta >0$ such that for all $x,y\in A$ $|x-y|<\delta \implies |f(x)-f(y)| < \varepsilon . $ > [!Example] Example (Uniformly continuous function) > ![[Screenshot 2024-10-16 at 15.08.27.png]] > (from wikipedia). # Properties In general, [[Uniformly Continuous Real Function Grows At Most Linearly]]. # Application(s) Note that continuous reals function on closed intervals [[Continuous Real Function on Closed Real Interval is Uniformly Continuous|are uniformly continuous]]. # Source(s) 1. https://en.wikipedia.org/wiki/Uniform_continuity