# Definition:Differentiable Mapping/Complex Function

## Definition

### At a Point

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

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

Let $z_0 \in D$ be a point in $D$.

Then $f$ is complex-differentiable at $z_0$ if and only if the limit:

$\displaystyle \lim_{h \to 0} \frac {f \left({z_0+h}\right) - f \left({z_0}\right)} h$

exists as a finite number.

### In an Open Set

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 $D$.