In mathematical logic

Mathematical logic

Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...

, specifically computability theory

Recursion theory

Computability theory, also called recursion theory, is a branch of mathematical logic that originated in the 1930s with the study of computable functions and Turing degrees. The field has grown to include the study of generalized computability and definability...

, a function

Range

-In mathematics:* Range , the set of all output values produced by a function* Interval , also called a range, a set of real numbers that includes all numbers between any two numbers in the set...

  is sequentially computable if, for every computable sequence  of real number

Real number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s, the sequence

Sequence

In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...

  is also computable.

A function is effectively uniformly continuous if there exists a recursive function

Primitive recursive function

The primitive recursive functions are defined using primitive recursion and composition as central operations and are a strict subset of the total µ-recursive functions...

  such that, if



then



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

These definition

Definition

A definition is a passage that explains the meaning of a term , or a type of thing. The term to be defined is the definiendum. A term may have many different senses or meanings...

s can be generalized to functions of more than one variable

Variable (mathematics)

In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...

 or functions only defined on a subset

Subset

In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...

of The generalizations of the latter two need not be restated. A suitable generalization of the first definition is:

Let be a subset of A function is sequentially computable if, for every -tuplet of computable sequences of real numbers such that



the sequence is also computable.