Computable real function

In mathematical logic, specifically computability theory, a function $f:\mathbb {R} \to \mathbb {R}$ 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 $f:\mathbb {R} \to \mathbb {R}$ is effectively uniformly continuous if there exists a primitive recursive function $d:\mathbb {N} \to \mathbb {N}$ such that, if $|x-y|<1/d(n)$, then $|f(x)-f(y)|<1/n$.

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

These definitions can be generalized to functions of several variables as follows: let $D$ be a subset of $\mathbb {R} ^{n}$. A function $f:D\to \mathbb {R}$ is sequentially computable if, for every $n$-tuple $(\{x_{i\,1}\}_{i=1}^{\infty },\dots ,\{x_{i\,n}\}_{i=1}^{\infty })$ of computable sequences of real numbers such that

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

for all $i$, the sequence $\{f(x_{i})\}_{i=1}^{\infty }$ is also computable.