Lipschitz Continuity

analysis

Definition

Lipschitz Continuity

A function $f:D\to \mathbb{R}$ is called Lipschitz continuous if there exists a real constant $L\ge 0$ such that, for all $x_1,x_2\in D$:

$$|f(x_1)-f(x_2)|\le L|x_1-x_2|$$

The constant $L$ is referred to as the Lipschitz constant. Intuitively, a Lipschitz continuous function is limited in how fast it can change: the slope of the secant line between any two points on the graph never exceeds $L$ in absolute value.

Properties

• Implication of Continuity: Every Lipschitz continuous function is uniformly continuous, and therefore continuous. The converse is not necessarily true (e.g., $f(x)=\sqrt{x}$ is not Lipschitz continuous at $x=0$).
• Bounded Derivatives: A differentiable function is Lipschitz continuous if and only if its derivative is bounded.
• Rademacher’s Theorem: A Lipschitz continuous function is differentiable almost everywhere.