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Nestoridis","submitted_at":"2017-09-01T12:45:07Z","abstract_excerpt":"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. 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