A Lipschitz function is uniformly continuous.
We frequently encounter a particularly strong form of Lipschitz continuity where the function maps a metric space into itself with modulus less than one. Such a function, which maps points closer together, is called a contraction. Specifically, an operator f: X → X is called a contraction mapping if (or simply a contraction) if there exists a constant β, 0 ≤ P < 1, such that
p(f (x), f (x0)) < βp(x, x0)
for every x, x0 ∊ X. Contraction mappings are valuable in economic analysis since they can easily be shown to have a unique fixed point (theorem 2.5).