Non-isolated singularity

A singularity is non-isolated when every punctured neighbourhood contains another singularity. An accumulation point of poles or essential singularities cannot itself be classified as a removable singularity, pole, or isolated essential singularity.

The function $1/\cos (1/z)$ has simple poles at $z=1/[\pi (n+1/2)]$, accumulating at zero. Its origin is therefore a non-isolated singularity, despite the underlying cosine of $1/z$ having an isolated essential singularity.