A global Lipschitz bound gives one delta that works everywhere.Suggest a slogan
Let \((X,d_X)\) and \((Y,d_Y)\) be metric spaces and let \(f\colon X\to Y\) be Lipschitz: there is \(C>0\) with \(d_Y(f(x),f(y))\leq C\,d_X(x,y)\) for all \(x,y\in X\). Then \(f\) is uniformly continuous.
Given \(\varepsilon>0\), take \(\delta\coloneqq\varepsilon/C\). If \(d_X(x,y)<\delta\), then \[ d_Y(f(x),f(y)) \leq C\,d_X(x,y) < C\delta = \varepsilon . \]