lipschitzconstante

Lipschitzconstante, in mathematics commonly called the Lipschitz constant, is a nonnegative real number that bounds how rapidly a function can change with respect to its input. It provides a quantitative measure of the function's regularity in a metric space.

Let f: X → Y be a function between metric spaces (X, d_X) and (Y, d_Y). If there exists L ≥ 0 such that d_Y(f(x1), f(x2)) ≤ L d_X(x1, x2) for all x1, x2 ∈ X, then f is called Lipschitz, and L is a Lipschitz constant for f. If such an L exists, the smallest one is the Lipschitz constant. If no finite L works, the function is not Lipschitz.

Common contexts: in Euclidean spaces R^n with the usual metric, many functions have a Lipschitz constant given by bounds on derivatives. For differentiable f on a convex set with sup norm of the derivative ‖Df‖ ≤ M, f is M-Lipschitz. For a linear map A, the operator norm ‖A‖ is a Lipschitz constant.

Examples: f(x) = ax + b on R is |a|-Lipschitz. f(x) = |x| is 1-Lipschitz. The projection onto a closed convex set is 1-Lipschitz. The distance function x ↦ d(x, p) is 1-Lipschitz.

Properties: Lipschitz functions are uniformly continuous. The composition of an L-Lipschitz map with an M-Lipschitz map is LM-Lipschitz. The sum of two Lipschitz maps with constants L1 and L2 is (L1+L2)-Lipschitz. Local versus global: a function can be locally Lipschitz (L may depend on the neighborhood) or globally Lipschitz (a single L for all inputs in the domain).