cv.complex variables - Example of continuous function that is analytic on the interior but cannot be analytically continued?

I found some neat stuff in Remmert's Classical topics in complex function theory.

Fabry's gap theorem gives a way to construct many examples including some already mentioned. Stated for the unit disk, it says:

If $m_1,m_2,ldots$ is a sequence of positive integers such that $displaystyle{lim_{ntoinfty}}frac{m_n}{n}=infty$ and if $displaystyle{f(z)=sum_{n=1}^infty a_nz^{m_n}}$ has radius of convergence 1, then the unit disk is the domain of holomorphy of $f$.

For example, if $p_n$ is the $n^{th}$ prime, then

$$f(z)=sum_{n=1}^infty frac{z^{p_n}}{n^2}$$ converges uniformly on the closed disk and is therefore continuous. It is not analytically extendable to any larger set because it satisfies the hypotheses of Fabry's theorem.

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An interesting result that yields many such functions in a nonconstructive way is a theorem of Fatou-Hurwitz-Pólya:

If $displaystyle{f(z)=sum_{n=0}^infty a_n z^n}$ has radius of convergence 1, then the set of functions

$$f_epsilon(z)=sum_{n=0}^infty epsilon_na_nz^n$$ for $epsilon_nin{pm1}$ whose domain of holomorphy is the unit disk has cardinality $2^{aleph_0}$.

Hausdorff showed further that if $displaystyle{lim_{ntoinfty} |a_n|^{1/n}}$ exists (and equals 1) then the set of such functions whose domain of holomorphy is not the unit disk is at most countable. This applies in particular to the function $displaystyle{f(z)=sum_{n=1}^infty frac{z^n}{n^2}}$, which therefore yields examples by changing the signs of the coefficients in all but countably many ways.

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One more, this time an explicit example from Remmert: The series

$$f(z)=1+2z+sum_{n=1}^inftyfrac{z^{2^n}}{2^{n^2}}$$ is one-to-one and has real derivatives of all orders on the closed disk, and has the open disk as domain of holomorphy.

Reference: Remmert's Classical topics in complex function theory, pages 252-258. (Fatou-Hurwitz-Pólya is stated on a page without preview.)

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