It is well-known that the function $f(z)=\sum\_{n=0}^\infty z^{n!}$ is analytic in the open unit disk and it can not be extended analytically to any proper open superset of the unit disk, i.e., the unit circle is the natural boundary of $f$.


Is there an example of a function, analytical in the unit disk, with "natural boundary" a part (of positive 1-dimensional measure) of the unit circle, which extends analytically to rest of $\mathbb C$. 


I am not asking for a variation of $g(z)=\sqrt{z^2-1}$, which is defined in $\mathbb C\smallsetminus [-1,1]$, since the segment $[-1,1]$ is not a "natural boundary" of $g$, as $g$ can be analytically continued through this segment.



Can the natural boundary be part of the unit circle?
Every closed set $F$ on the unit circle is the set of singularities of some analytic function.
Take a countable dense subset $z\_k$ of $F$ and then choose positive $a\_k$ so small
that the series
$$f(z)=\sum\_k\frac{a\_k}{z-z\_k}$$
converges uniformly on compact subsets of $C\backslash F$.


