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:

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

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

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