# Definition:Differentiable Mapping/Complex Function/Region

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## Definition

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.

## Sources

- 2001: Christian Berg:
*Kompleks funktionsteori*: $\S 1.1$