  Kleene's computability theory based on the S1-S9 computation schemes
constitutes a model for computing with objects of any finite type and extends
Turing's 'machine model' which formalises computing with real numbers. A
fundamental distinction in Kleene's framework is between normal and non-normal
functionals where the former compute the associated Kleene quantifier
$\exists^n$ and the latter do not. Historically, the focus was on normal
functionals, but recently new non-normal functionals have been studied based on
well-known theorems, the weakest among which seems to be the uncountability of
the reals. These new non-normal functionals are fundamentally different from
historical examples like Tait's fan functional: the latter is computable from
$\exists^2$, while the former are computable in $\exists^3$ but not in weaker
oracles. Of course, there is a great divide or abyss separating $\exists^2$ and
$\exists^3$ and we identify slight variations of our new non-normal functionals
that are again computable in $\exists^2$, i.e. fall on different sides of this
abyss. Our examples are based on mainstream mathematical notions, like
quasi-continuity, Baire classes, bounded variation, and semi-continuity from
real analysis.
