Definition:Derivative/Complex Function/Open Set

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Let $D\subseteq \C$ be an open set.

Let $f : D \to \C$ be a complex function.

Let $f$ be complex-differentiable in $D$.

Then the derivative of $f$ is the complex function $f': D \to \C$ whose value at each point $z \in D$ is the derivative $\map {f'} z$:

$\ds \forall z \in D : \map {f'} z := \lim_{h \mathop \to 0} \frac {\map f {z + h} - \map f z} h$