The Complex Derivative at a Point

In the setting of Complex Analysis: Standing Notation, let $U$ be an open subset of the metric space $(\mathbb{C},d_{\mathbb{C}})$, let $f:U\to\mathbb{C}$ be a function, and let $z\in U$.

(i) Complex derivative at a point. A complex number $L$ is a complex derivative of $f$ at $z$ if for every real $\varepsilon>0$ there is a real $\delta>0$ such that every $w\in U$ with $0<|w-z|<\delta$ satisfies

$$\Bigl|\frac{f(w)-f(z)}{w-z}-L\Bigr|<\varepsilon ,$$

the quotient being defined because $w\neq z$.

(ii) Complex differentiability. $f$ is complex differentiable at $z$ if it has a complex derivative at $z$. This complex derivative is then unique, by Uniqueness of the Complex Derivative (i), and is written $f'(z)$.