Lipschitz

Lipschitz refers to a mathematical concept named after the German mathematician Rudolf Lipschitz. In analysis and geometry, Lipschitz conditions place a uniform bound on how rapidly a function can change its output relative to changes in its input.

A function f from a metric space (X, d_X) to another metric space (Y, d_Y) is called L-Lipschitz if there exists a constant L ≥ 0 such that for all x, y in X, d_Y(f(x), f(y)) ≤ L d_X(x, y). If such an L can be chosen for all points in X, f is globally Lipschitz; if the inequality holds only in a neighborhood of each point (with a possibly different L), f is locally Lipschitz. Lipschitz maps are automatically uniformly continuous, and their behavior is stable under composition: if f is L-Lipschitz and g is M-Lipschitz, then g∘f is (LM)-Lipschitz.

In Euclidean spaces, a common form is |f(x) − f(y)| ≤ L||x − y|| for all x, y. Sufficient conditions for Lipschitz continuity include a differentiable function with bounded derivative on a convex domain; then the supremum of the derivative bounds the Lipschitz constant. Examples include linear maps, which are Lipschitz with constant equal to the operator norm, and the sine function, which is 1-Lipschitz on the real line. The function f(x) = √x on [0, ∞) is not globally Lipschitz, though it is uniformly continuous.

Variants of the concept include bi-Lipschitz maps (where both f and its inverse are Lipschitz) and Hölder continuity (a weaker condition). Beyond real analysis, Lipschitz continuity is central in differential equations, optimization, geometry, and the study of metric spaces.