Hierarchies of Function Classes Defined by the First-Value Operator

In this paper, the first-value operator is applied to establish hierarchies of classes of functions under various settings. For effective sequences of computable discrete functions, we obtain a hierarchy connected with Ershov's one within Δ20. The non-effective version over real functions is connected with the degrees of discontinuity and yields a hierarchy related to Hausdorff's difference hierarchy in the Borel class Δ2B. Finally, the effective version over approximately computable real functions forms a hierarchy which provides a useful tool in computable analysis.