1 Calculus, Continuity, and Computability 3 1.1 When to Introduce Relative Computability? . . . . . . . . . . 4 1.2 Between Computability and Relative Computability? . . . . . 5 1.3 The Development of Relative Computability . . . . . . . . . . 5 1.4 Turing Introduces Relative Computability . . . . . . . . . . . 6 1.5 Post Develops Relative Computability . . . . . . . . . . . . . 6 1.6 Relative ...

We consider real sequences in I = [0, 1) and real functions on I . It is first shown that, as for real sequences from I , R-computability (computability with respect to the Euclidean topology) implies “weak Fine-computability.” Using this result, we show that “Fine-sequential computability” and “L∗-sequential computability” are equivalent for effectively locally Fine-continuous functions as wel...

In classical computability theory the main underlying structure is that of the natural numbers or equivalently a structure consisting of some constructive objects, such as words in a finite alphabet. In the 1960’s computability theorists saw it as a challenge to extend the notion of computable to arbitrary structure. The resulting subfield of computability theory is commonly referred to as comp...