Localized versions of function spaces and generic results

We consider generalizations of classical function spaces by requiring that a holomorphic in $\mathrm{\Omega }$ function satisfies some property when we approach from $\mathrm{\Omega }$, not the whole boundary, but only a part of it. These spaces endowed with their natural topology are Fr$\acute{e}$chet spaces. We prove some generic non-extendability results in such spaces and generic nowhere differentiability on the corresponding part of $\mathrm{\partial }\mathrm{\Omega }$.