# Complex Differentiable Functions

We will now touch upon one of the core concepts in complex analysis - differentiability of complex functions baring in mind that the concept of differentiability of a complex function is analogous to that of a real function.

Definition: Let $A \subseteq C$ be open, $z_0 \in A$, and let $f : A \to \mathbb{C}$. Then $f$ is said to be Complex Differentiable at $z_0$ if the limit $\displaystyle{f'(z_0) = \lim_{h \to 0} \frac{f(z_0 + h) - f(z_0)}{h}}$ exists. The limit, $f'(z_0)$ is called the Derivative of $f$ at $z_0$ provided that this limit exists. |

*Note that if $h = z - z_0$ then $h \to 0$ if and only if $z \to z_0$, in which case we have that:*

*Also note that in both limit definitions above - the limit exists if and only if the limit is finite and the same for all paths in $\mathbb{C}$ as $h \to 0$ (or as $z \to z_0$).*

For a simple example, consider the following function:

(2)We will show that this function is differentiable at every $z_0 \in \mathbb{C}$. We have that:

(3)Therefore $f$ is differentiable at every point $z_0 \in \mathbb{C}$ and the derivative of $f$ at every $z_0 \in \mathbb{C}$ is $f'(z_0) = 1$.

For another example, consider the following function:

(4)We will show that this function is differentiable at $0$. We have that:

(5)The following theorem tells us that if a function is complex differentiable at a point then it is continuous at that point.

Theorem 1: Let $A \subseteq \mathbb{C}$ be open, $z_0 \in A$, and $f : A \to \mathbb{C}$. If $f$ is complex differentiable at $z_0$ then $f$ is continuous at $z_0$. |

**Proof:**Since $f$ is complex differentiable at $z_0$, $f'(z_0)$ exists and so:

- Therefore $\displaystyle{\lim_{z \to z_0} f(z) = f(z_0)}$ and so $f$ is continuous at $z_0$. $\blacksquare$