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