Let \((X, d_1)\) and \((Y, d_2)\) be metric spaces. A function \(f\colon X \to Y\) is uniformly continuous if for every \(\varepsilon > 0\) there exists \(\delta=\delta(\varepsilon) > 0\) such that for all \(x,y\in X\), \[ x\in B_\delta(y) \implies f(x) \in B_\varepsilon(f(y)). \]
For continuity of \(f\) at every point, the same condition is required with \(\delta=\delta(\varepsilon,y)\) allowed to depend on the point \(y\); uniform continuity requires one \(\delta\) for all \(y\in X\).
What does it mean for $f: (X, d_1) \to (Y, d_2)$ to be uniformly continuous?