Definition:Sequentially Computable 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 that, for every $n$-tuple of infinite sequences:
- $\tuple {\sequence {x_{1,k}}_{k \mathop \in \N}, \sequence {x_{2,k}}_{k \mathop \in \N}, \dotsc, \sequence {x_{n,k}}_{k \mathop \in \N}}$
such that both of the following hold:
- $\paren 1 \quad$ For every $k \in \N$, $\tuple {x_{1,k}, x_{2,k}, \dotsc, x_{n,k}} \in D$.
- $\paren 2 \quad$ Each $\sequence {x_{i,k}}_{k \in \N}$ is a computable real sequence.
it necessarily holds that:
- $\sequence {\map f {x_{1,k}, x_{2,k}, \dotsc, x_{n,k}}}_{k \mathop \in \N}$ is a computable real sequence.
Then, $f$ is sequentially computable.
Sources
- This article incorporates material from computable real function on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.