A function $f : \mathbb{R}^n \to \mathbb{R}^m$ is Lipschitz continuous on a domain $D \subseteq \mathbb{R}^n$ if there exists a constant $L \geq 0$ such that

for all $x,y \in D$.

Here:

• $f$ is the function.
• $\vec{x},\vec{y} \in D$ are input vectors.
• $L \in \mathbb{R}_{\geq 0}$ is a Lipschitz constant.
• $\|\cdot\|$ denotes a norm, typically the Euclidean norm.

Intuitively, Lipschitz continuity means that the change in the function's output is bounded by a constant multiple of the change in its input.

For a scalar-valued function of one variable, Lipschitz continuity means that the absolute slope between any two points is bounded by $L$.

A holomorphic function is a function of a complex variable that is complex differentiable at every point in some open region of the complex plane.