Holomorphic function
Let $f$ be a complex valued function of a single variable.
On an open set: Definition. A function $f(z)$ is holomorphic on an open set $U$ if it is (complex-)differentiable at every pointin $U$.
On a non-open set: Definition. A function $f(z)$ is holomorphic on a non-open set $A$ if it is holomorphic on some open set containing $A$.
At a point: Definition. A function $f(z)$ is holomorphic at $z_o$ if it is differentiable on some neighborhood of $z_o$.