Computability of a function with jumps*1Effective uniformity and limiting recursion

Inside the Muchnik degrees I: Discontinuity, learnability and constructivism

“…Although it seems easy to draw a rough graph of the floor function, it is not computable with respect to the standard real number representation [77], because computability automatically induces topological continuity. One way to study the floor function in computability theory is to "computabilize" it by changing the representation/topology of the real space (see, for instance, [79]). However, it is also important to enhance our knowledge of the noncomputability/discontinuity level of such seemingly computable functions without changing representation/topology.…”

“…There have been many challenges [15,82,81,80,83,78,79] in developing computability theory of (nonuniformly computable) discontinuous functions using the notion of learnability (dynamicalapproximation) and piecewise computability (statical-approximation). Indeed, one can show the equivalence of effective learnability and Π 0 1 -piecewise computability: the class of functions that are computable with finitely many mind changes is exactly the class of functions that are decomposable into countably many computable functions with Π 0 1 domains.…”