On the computational properties of basic mathematical notions

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