Analytic

Complex-differentiable at every point of some open set around $z_0$ , not just at $z_0$ alone.

A function $f$ is analytic at $z_0$ if it has a complex derivative at every point of some open neighborhood of $z_0$ -- differentiability at the single point $z_0$ is not enough. This 'differentiable throughout a whole neighborhood' requirement is what gives analytic functions their unusually strong consequences (infinite differentiability, power series representations, and more), covered later in this course. The terms analytic and holomorphic are used interchangeably throughout this course.

Examples

$f(z)=z^2$ is analytic at every point of the complex plane, since it's differentiable on an open neighborhood of every point.
A function can be differentiable at one isolated point without being analytic there: $f(z)=|z|^2$ is differentiable only at $z=0$ , hence analytic nowhere, since no full neighborhood of $0$ consists entirely of differentiable points.