# computable real function

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

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

$$ |

then

$$ |

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 $D$ be a subset of ${\mathbb{R}}^{n}$. A function $f:D\to \mathbb{R}$ is *sequentially computable* if, for every $n$-tuplet $({\{{x}_{i\mathrm{\hspace{0.17em}1}}\}}_{i=1}^{\mathrm{\infty}},\mathrm{\dots}{\{{x}_{in}\}}_{i=1}^{\mathrm{\infty}})$ of computable sequences of real numbers such that

$$(\forall i)\mathit{\hspace{1em}}({x}_{i\mathrm{\hspace{0.17em}1}},\mathrm{\dots}{x}_{in})\in D\mathit{\hspace{1em}\hspace{1em}},$$ |

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

Title | computable real function |
---|---|

Canonical name | ComputableRealFunction |

Date of creation | 2013-03-22 14:39:23 |

Last modified on | 2013-03-22 14:39:23 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 6 |

Author | rspuzio (6075) |

Entry type | Definition |

Classification | msc 03F60 |

Defines | sequentially computable |

Defines | effectively uniformly continuous |

Defines | effective uniform continuity |