In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for function (mathematics)|functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is no greater than a definite real number; this bound is called the function's Lipschitz constant (or modulus of continuity|modulus of uniform continuity).
In the theory of differential equations, Lipschitz continuity is the central condition of the Picard–Lindelöf theorem which guarantees the existence and uniqueness of the solution to an initial value problem. A special type of Lipschitz continuity, called contraction mapping|contraction, is used in the Banach fixed point theorem.
The concept of Lipschitz continuity is well-defined on metric spaces. A generalization of Lipschitz continuity is called Hölder continuity.