Lipschitz function,
n. a function between normed spaces with the property that the distance between function values is bounded by a constant multiple of the distance between the arguments. If the function satisfies the Lipschitz condition that
|| f(x) - f(y) || ≤ k || x - y ||
for all x and y in a set A or at a point x0 then f is k-Lipschitz on A or at x0. For example, f(x) = x2 is 2-Lipschitz on (-1, 1), since
| x2 - y2| = | x + y| | x - y| ≤ 2 | x - y|.
When k = 1, the function is non-expansive, and is a contraction if k < 1. The Rademacher theorem proves that every finite-dimensional Lipschitz function is differentiable almost everywhere. More generally, if a function satisfies the Lipschitz condition of order p (also known as the Hölder condition) that
|| f(x) - f(y) || ≤ k || x - y || p
for some 0 < p ≤ 1, for x and y in a set A, then f is said to be Hölder-continuous on A. (Named after the German analyst, algebraist, number theorist and physicist, Rudolph Otto Sigismund Lipschitz (1832-1903).)
|| f(x) - f(y) || ≤ k || x - y ||
for all x and y in a set A or at a point x0 then f is k-Lipschitz on A or at x0. For example, f(x) = x2 is 2-Lipschitz on (-1, 1), since
| x2 - y2| = | x + y| | x - y| ≤ 2 | x - y|.
When k = 1, the function is non-expansive, and is a contraction if k < 1. The Rademacher theorem proves that every finite-dimensional Lipschitz function is differentiable almost everywhere. More generally, if a function satisfies the Lipschitz condition of order p (also known as the Hölder condition) that
|| f(x) - f(y) || ≤ k || x - y || p
for some 0 < p ≤ 1, for x and y in a set A, then f is said to be Hölder-continuous on A. (Named after the German analyst, algebraist, number theorist and physicist, Rudolph Otto Sigismund Lipschitz (1832-1903).)