Computable real function

In mathematical logic, specifically computability theory, 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

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

then

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

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

These definitions can be generalized to functions of more than one variable or functions only defined on a subset of $\mathbb{R}^n.$ The generalizations of the latter two need not be restated. A suitable generalization of the first definition is:

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

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

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

This article incorporates material from Computable real function on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.