= Natural boundary of a holomorphic function
= Natural boundary
{synonym}
A boundary of a domain of a <holomorphic function> is natural if the function has no analytic continuation through any of its points. For $f(z)=\sum_{n\ge1}z^{2^n}$, each root of unity of order a power of two is a singular boundary point: along $z=r\zeta$ all sufficiently late terms equal $r^{2^n}$ and their sum tends to infinity as $r\uparrow1$. These roots are dense on the unit circle, so no boundary arc admits analytic continuation.
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