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 $\mathbb {C}$ and $f:U\to \mathbb {C}$ 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:

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

In subscript notation, they read more compactly as:

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