Definition:Uniformly Equicontinuous

Definition

Let $X = \struct {A, d}$ and $Y = \struct {B, \rho}$ be metric spaces.

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

Then $\family {f_i}_{i \mathop \in I}$ is said to be uniformly equicontinuous on $S \subseteq A$ if and only if:

$\forall \epsilon \in \R_{>0} : \exists \delta \in \R_{>0}: \forall i \in I: \forall x, y \in S : \map d {x, y} < \delta \implies \map \rho {\map {f_i} x, \map {f_i} y} < \epsilon$

Also see

• Results about uniform equicontinuity can be found here.

Sources

• 2003: Charles C. Pugh: Real Mathematical Analysis (2nd ed.) ... (next): $\S 4.3$