Function real-differentiable at a point

Definition

For an open subset of the complex numbers

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 $f$ is real-differentiable at $z_0$ if there exists a matrix $(Df)\in M_2(\mathbb{R})$ such that:

$\underset{h\to 0}{lim}\frac{f(z+h)-f(z)-(Df)(h)}{\left|h\right|}=0$

Here $(Df)(h)$ denotes $Df$ multiplied with $h$ as a column vector.

Such a $Df$ is termed a real differential of $f$.