Function satisfying Cauchy-Riemann differential equations at a point

This article defines a property that can be evaluated for a function on a (particular kind of) set, and a point in that set. A function satisfying the property at every point, it is termed a ?
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Definition

Suppose $U$ is an open subset of $C$ and $f : U \to C$ is a function. Let $z_{0} \in U$ be a point. We say that $f$ satisfies the Cauchy-Riemann differential equations at $z_{0}$ if the following is true:

Both the real and the imaginary part of $f$ have well-defined partial derivatives in the real and imaginary direction, at the point $z_{0}$
The partial derivatives satisfy the Cauchy-Riemann differential equations. If we denote by $u, v$ the real and imaginary parts of $f$ , the Cauchy-Riemann differential equations state that:

$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x}$

In subscript notation, they read more compactly as:

$u_{x} = v_{y}, u_{y} = - v_{x}$