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Baire category and nowhere differentiability for feasible real functions

A notion of resource‐bounded Baire category is developed for the class PC[0,1] of all polynomial‐time computable real‐valued functions on the unit interval. The meager subsets of PC[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 PC[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. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)