Definition 3.4.1.
Let \(S \subset \R\text{,}\) and let \(f \colon S \to \R\) be a function. Suppose for every \(\epsilon > 0\) there exists a \(\delta > 0\) such that whenever \(x, c \in S\) and \(\abs{x-c} < \delta\text{,}\) then \(\abs{f(x)-f(c)} < \epsilon\text{.}\) Then we say \(f\) is uniformly continuous.