 Clearly "f" has a singularity at "z" = 1, because

is a divergent series.But if "z" is allowed to be non-real, problems arise, since

we can see that "f" has a singularity at a point "z" when "z"a = 1, and also when "z"a2 = 1.By the induction suggested by the above equations, "f" must have a singularity at each of the "a""n"-th roots of unity for all natural numbers "n." The set of all such points is dense on the unit circle, hence by continuous extension every point on the unit circle must be a singularity of "f."

Evidently the argument advanced in the simple example shows that certain series can be constructed to define lacunary functions.