Removable singularity

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In complex analysis, a removable singularity of a function is a point at which the function is not defined (a singularity) but at which the function can be so defined that it is continuous at the singularity.

For instance, the function

$f(z) = \frac{\sin z}{z}$

for z ≠ 0 has a removable singularity at z = 0: we can define f(0) = 1 and the resulting function will be continuous and even infinitely differentiable (a consequence of L'Hôpital's rule).

Formally, if U is an open subset of the complex plane C, a is an element of U and f : U - {a} → C is a holomorphic function, then a is called a removable singularity for f if there exists a holomorphic function g : U → C which coincides with f on U - {a}. Such a holomorphic function g exists if and only if the limit lim_z→a f(z) exists; this limit is then equal to g(a).

Riemann's theorem on removable singularities states that a singularity a of a holomorphic function f is removable if and only if there exists a neighborhood of a on which f is bounded.

The removable singularities are precisely the poles of order 0. Equivalently, $a$ is removable if and only if $\lim_{z\rightarrow a} (z-a) f(z) = 0.$