# Removable singularity

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*) = 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 differentiable (a consequence of L'Hopital'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 *z* 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 the singularity *a* 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.