Definition:Computably Uniformly Continuous Real-Valued Function

Definition

Let $D \subseteq \R^n$ be a subset of real cartesian $n$-space.

Let $f : D \to \R$ be a real-valued function on $D$.

Suppose there exists a total recursive function $d : \N \to \N$ such that:

For every $n \in \N$ and $\bsx, \bsy \in D$ such that:

$\norm {\bsx - \bsy} < \dfrac 1 {\map d n + 1}$

where $\norm \cdot$ is the Euclidean norm, it holds that:

$\size {\map f \bsx - \map f \bsy} < \dfrac 1 {n + 1}$

Then $f$ is computably uniformly continuous.

Sources

This article incorporates material from computable real function on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.