TITLE: What type of singularity $\sin(\frac{1}{\cos(\frac{1}{z})})$ has? (GATE 2009)
QUESTION [9 upvotes]: Determine the type of singularity at $z = 0$ of the following function and why?

$$f(z) = \sin \left(\frac{1}{\cos (\frac{1}{z})}\right).$$
I have no idea. 
Thank you for your answer.

REPLY [7 votes]: The singularity at $z=0$ is not isolated (since $\cos(1/z) = 0$ for a sequence of points tending to $z=0$). The usual classification into removable singularities, poles and essential singularities is only applicable to isolated singularities.
(Your textbook may have another convention.)