Baire category and nowhere differentiability for feasible real functions

Abstract

A notion of resource-bounded Baire category is developed for the class P C[0,1] of all polynomial-time computable real-valued functions on the unit interval. The meager subsets of P C[0,1] are characterized in terms of resource-bounded Banach-Mazur games. This characterization is used to prove that, in the sense of Baire category, almost every function in P C[0,1] is nowhere differentiable. This is a complexity-theoretic extension of the analogous classical result that Banach proved for the class C[0, 1] in 1931. © 2001 Springer Berlin Heidelberg.