Definition:Computably Uniformly Continuous Real Function

Definition

Let $f : \R \to \R$ be a real function.

Suppose there exists a total recursive function $d : \N \to \N$ such that:

For every $n \in \N$ and $x, y \in \R$ such that:

$\size {x - y} < \dfrac 1 {\map d n + 1}$

it holds that:

$\size {\map f x - \map f y} < \dfrac 1 {n + 1}$

Then $f$ is computably uniformly continuous.

Also known as

This property is also called effectively uniformly continuous, due to the Church-Turing Thesis.

Sources

• 1957: A. Grzegorczyk: On the definitions of computable real continuous functions (Fund. Math. Vol. 44, no. 1: pp. 61 – 71)