In mathematics, a function *f* : *M* → *N* between metric spaces *M* and *N* is called **Lipschitz continuous** (or is said to satisfy a **Lipschitz condition**) if there exists a constant *K* > 0 such that d(*f*(*x*), *f*(*y*)) ≤ *K* d(*x*, *y*) for all *x* and *y* in *M*. In this case, *K* is called the **Lipschitz constant** of the map. The name comes from the German mathematician Rudolf Lipschitz.

Every Lipschitz continuous map is uniformly continuous and hence continuous.

Lipschitz continuous maps with Lipschitz contant *K* < 1 are called *contraction mappings*; they are the subject of the Banach fixed point theorem.

Lipschitz continuity is an important condition in the existence and uniqueness theorem for ordinary differential equations.

If *U* is a subset of the metric space *M* and *f* : *U* → **R** is a real-valued Lipschitz continuous map, then there always exist Lipschitz continuous maps *M* → **R** which extend *f* and have the same Lipschitz constant as *f*.

A Lipschitz continuous map *f* : *I* → **R**, where *I* is an interval in **R**, is almost everywhere differentiable (everywhere except on a set of Lebesgue measure zero). If *K* is the Lipschitz constant of *f*, then |*f'*(*x*)| ≤ *K* whenever the derivative exists. Conversely, if *f* : *I* → **R** is a differentiable map with bounded derivative, |*f'*(*x*)| ≤ *L* for all *x* in *I*, then *f* is Lipschitz continuous with Lipschitz constant *K* ≤ *L*, a consequence of the mean value theorem.