On the computational properties of ambivalent sets and functions

Abstract

Examples of discontinuous functions already appear in the work of Euler, Abel, Dirichlet, Fourier, and Bolzano. A ground-breaking discovery due to Baire was that many discontinuous functions are well-behaved in that they are the pointwise limit of a sequence of continuous functions; the latter form a class nowadays simply called `Baire 1'. We shall study a class strictly between the semi-continuous and Baire 1 functions, called the ambivalent fuctions. In particular, we investigate the computational properties of the class of ambivalent functions and sets, denoted $\bf \Delta$, working with Kleene's S1-S9 schemes. Computational equivalences for various standard operations (supremum, Baire 1 representation, \dots) on $\bf \Delta$ are established, including the structure functional $\Omega_{\bf \Delta}$ that decides if a given ambivalent set is non-empty. A selector is shown to be computable relative to $\Omega_{\bf \Delta}$ and Kleene's quantifier $\exists^{2}$.