  Kleene's computability theory based on his 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 like 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 only computable in $\exists^{3}$. While there is a great divide
separating $\exists^{2}$ and $\exists^{3}$, we identify certain closely related
non-normal functionals that fall on different sides of this abyss. Our examples
are based on mainstream mathematical notions, like quasi-continuity, Baire
classes, and semi-continuity.
