Differentiability

A complex function $f$ is differentiable at $z_0$ iff:

$f'(z_0)$ is called the derivative of $f$ at $z_0$. Properties of Differentiability | Real Analysis can be applied to complex functions.

Singular Point

A point $z_0$ where $f(z)$ is not differentiable.

Neighbourhood

Suppose $z_0 \in \mathbb{C}$. A neighborhood of $z_0$ is the region contained in the circle $|z - z_0| = r \gt 0$.