f(z) is nowhere analytic.

There exists z_0\in\mathbb{C} such that f(z) is continuous at z_0.

None of the others.

There exists z_0\in\mathbb{C} such that f(z) is differentiable at z_0.

f(z) is discontinuous at 0.

There exists z_0\in\mathbb{C} such that f(z) is analytic at z_0.