Localized versions of function spaces and generic results

Abstract We consider generalizations of classical function spaces by requiring that a holomorphic in Ω function satisfies some property when we approach from Ω, not the whole boundary ∂Ω, but only a part of it. These spaces endowed with their natural topology are Frechet spaces. We prove some generic non-extendability results in such spaces and generic nowhere differentiability on the corresponding part of ∂Ω.