Definition:Uniformly Equicontinuous

Definition
Let $X = \left({A, d}\right)$ and $Y = \left({B, \rho}\right)$ be metric spaces.

Let $\left\langle{f_i}\right\rangle_{i \mathop \in I} $ be a family of mappings $f_i: X \to Y$.

Then $\left\langle{f_i}\right\rangle_{i \mathop \in I}$ is said to be uniformly equicontinuous on $S \subseteq A$ :
 * $\forall \epsilon \in \R_{>0} : \exists \delta \in \R_{>0}: \forall i \in I: \forall x, y \in S : d \left({x, y}\right) < \delta \implies \rho \left({f_i \left({x}\right), f_i \left({y}\right) }\right) < \epsilon$