Definition:Computably Uniformly Continuous Real-Valued Function
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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.