The answer is very easy: the function...
... is analytic on the whole complex plane...
I'm new to this stuff, but I wanted to check myself. I think it goes something like this:
clearly has a singularity at . I mean, it's undefined there so itself is not what is holomorphic, right? But because 0 is a removable singularity it is possible to define a function where the new function is everywhere equal to the original except at the singularity, where we can define to be the limit of as , which is legitimate because this limit exists and is finite. Have I understood this correctly?
Also, what is the justification that 0 is a removable singularity for this function? Is it, in fact, because exists and is finite? If so, would this need to be shown for completeness, or could it be taken as obvious enough?