Computable real function
In mathematical logic, specifically computability theory, a function is sequentially computable if, for every computable sequence of real numbers, the sequence is also computable.
A function is effectively uniformly continuous if there exists a recursive function such that, if
then
A real function is computable if it is both sequentially computable and effectively uniformly continuous,[1]
These definitions can be generalized to functions of more than one variable or functions only defined on a subset 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.
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References
- ^ see Grzegorczyk, Andrzej (1957), "On the Definitions of Computable Real Continuous Functions" (PDF), Fundamenta Mathematicae, 44: 61–77