Function complex-differentiable at a point

This article defines a property that can be evaluated for a function on a (particular kind of) set, and a point in that set. A function satisfying the property at every point, it is termed a holomorphic function
View other properties of functions at points

Definition

In one dimension

Let $U$ be a domain in $\mathbb {C}$ and $f:U\to \mathbb {C}$ be a function. Let $z_{0}\in U$. We say that $f$ is complex-differentiable at $z_{0}$ if the complex differential of $f$ at $z_{0}$ is well-defined.

Relation with other properties

Stronger properties

• Function holomorphic at a point

Weaker properties

• Function real-differentiable at a point
• Function satisfying Cauchy-Riemann differential equations at a point
• Function continuous at a point