Definition:Pointwise Equicontinuous

Definition

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

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

Then $\sequence {f_i}_{i \mathop \in I}$ is said to be pointwise equicontinuous at $x_0 \in A$ if and only if:

$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall i \in I: \forall x \in A: \map d {x, x_0} < \delta \implies \map \rho {\map {f_i} x, \map {f_i} {x_0} } < \epsilon$

Also see

• Results about pointwise equicontinuity can be found here.

Sources

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