  We consider the spaces $H_{F}^{\infty}(\Omega)$ and $\mathcal{A}_{F}(\Omega)$
containing all holomorphic functions $f$ on an open set $\Omega \subseteq
\mathbb{C}$, such that all derivatives $f^{(l)}$, $l\in F \subseteq
\mathbb{N}_0=\{ 0,1,...\}$, are bounded on $\Omega$, or continuously extendable
on $\overline{\Omega}$, respectively. We endow these spaces with their natural
topologies and they become Fr\'echet spaces. We prove that the set $S$ of
non-extendable functions in each of these spaces is either void, or dense and
$G_\delta$. We give examples where $S=\varnothing$ or not. Furthermore, we
examine cases where $F$ can be replaced by $\widetilde{F}=\{ l\in
\mathbb{N}_0:\min F \leqslant l \leqslant \sup F\}$, or $\widetilde{F}_0= \{
l\in \mathbb{N}_0:0\leqslant l \leqslant \sup F\}$ and the corresponding spaces
stay unchanged.
