Let $D=\{z\in \mathbb{C}\mid |z|<1\}$. Is there a holomorphic function $f:D\to \mathbb{C}$ such that for every $n\in \mathbb{N} \cup \{0\},\;f^{(n)}$ has a continuous extension to $\bar D$ but $f$ has no a holomorphic extension to any open neighborhood of $\bar D$?



Is there a holomorphic function on open unit disc with this property?
Yes, for example
$$f(z)=\sum\_{n=1}^\infty e^{-\log^2n}z^{n^2}.$$
The radius of convergence is $1$ by Hadamard's test. The function has gaps (the density of the sequence $n^2$ among the integers is zero) which are sufficient to
apply Fabry's gap theorem (or Polya's gap theorem) to ensure that this $f$ does not have any analytic continuation to a larger region than the unit disk,
and all derivatives are continuous in the closed disk because their power series are absolutely and uniformly convergent
in the closed disk.


Refs. It is difficult to find a good exposition of Fabry's work in English,
one can read his original papers (in French) or the exposition of Bieberbach (in his book Analytische Fortsetzung, in German). Polya work is also in German. But for the Polya gap theorem
one can consult Koosis, The Logarithmic integral, vol. II.


