PlanetMath: computable real function

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computable real function

(Definition)

A function $f \colon \mathbb{R} \to \mathbb{R}$ is sequentially computable if, for every computable sequence $\{x_i\}_{i=1}^\infty$ of real numbers, the sequence $\{f(x_i) \}_{i=1}^\infty$ is also computable.

A function $f \colon \mathbb{R} \to \mathbb{R}$ is effectively uniformly continuous if there exists a recursive function $d \colon \mathbb{N} \to \mathbb{N}$ such that, if

$\displaystyle \vert x-y\vert < {1 \over d(n)}$

then

$\displaystyle \vert f(x) - f(y)\vert < {1 \over n}$

A real function is computable if it is both sequentially computable and effectively uniformly continuous.

It is not hard to generalize these definitions to functions of more than one variable or functions only defined on a subset of $\mathbb{R}^n$ . The generalizations of the latter two definitions are so obvious that they need not be restated. A suitable generalization of the first definition is:

Let be a subset of $\mathbb{R}^n$ . A function $f \colon D \to \mathbb{R}$ is sequentially computable if, for every -tuplet $\left( \{ x_{i \, 1} \}_{i=1}^\infty, \ldots \{ x_{i \, n} \}_{i=1}^\infty \right)$ of computable sequences of real numbers such that

$\displaystyle (\forall i) \quad (x_{i \, 1}, \ldots x_{i \, n}) \in D \qquad ,$

the sequence $\{f(x_i) \}_{i=1}^\infty$ is also computable.

"computable real function" is owned by rspuzio.

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Also defines:

sequentially computable, effectively uniformly continuous, effective uniform continuity

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Cross-references: obvious, subset, variable, definitions, real function, recursive function, computable, sequence, real numbers, computable sequence, function

This is version 3 of computable real function, born on 2004-09-29, modified 2004-09-29.
Object id is 6248, canonical name is ComputableRealFunction.
Accessed 2740 times total.

Classification:

AMS MSC:

03F60 (Mathematical logic and foundations :: Proof theory and constructive mathematics :: Constructive and recursive analysis)

Pending Errata and Addenda

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