AbstractWe investigate the computational properties of basic mathematical notions pertaining to ${\mathbb R}\rightarrow {\mathbb R}$-functions and subsets of ${\mathbb R}$, like finiteness, countability, (absolute) continuity, bounded variation, suprema and regularity. We work in higher-order computability theory based on Kleene’s S1–S9 schemes. We show that the aforementioned italicised properties give rise to two huge and robust classes of computationally equivalent operations, the latter based on well-known theorems from the mainstream mathematics literature. As part of this endeavour, we develop an equivalent $\lambda $-calculus formulation of S1–S9 that accommodates partial objects. We show that the latter are essential to our enterprise via the study of countably based and partial functionals of type $3$.