Definition:Differentiable Mapping/Complex Function/Region

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

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

Then $f$ is complex-differentiable in $D$ if and only if $f$ is complex-differentiable at every point in $D$.

Also known as

A function which is complex-differentiable in an open set is called a holomorphic function in many texts.

It is also referred to as an analytic function.