Definition:Sequentially Computable 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 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.