Category:Definitions/Equicontinuous Real Functions

This category contains definitions related to Equicontinuous Real Functions.
Related results can be found in Category:Equicontinuous Real Functions.

Let $\family {f_i}_{i \mathop \in I}$ be an indexed family of real functions, where $I$ is an arbitrary index set.

Let each element of $\family {f_i}$ have the same domain.

Let $\family {f_i}$ have the property that:

$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \size {x_1 - x_2} < \delta \implies \size {\map {f_i} {x_1} - \map {f_i} {x_2} } < \epsilon$

for all $i \in I$.

Then $\family {f_i}_{i \mathop \in I}$ is a family of equicontinuous real functions.