Lipschitz continuity

In mathematical analysis, Lipschitz continuity is a regularity property of functions between metric spaces that is stronger than uniform continuity, and hence also stronger than continuity. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exists a real number such that, for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is not greater than this real number; the smallest such bound is called the Lipschitz constant of the function (and is related to the modulus of uniform continuity). For instance, every function that is defined on an interval and has a bounded first derivative is Lipschitz continuous.

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, is used in the Banach fixed-point theorem.

We have the following chain of strict inclusions for functions over a closed and bounded interval of the real line with non-empty interior:

Continuously differentiable ⊂ Lipschitz continuous ⊂

α

{\displaystyle \alpha }

-Hölder continuous,

where

0

<

α

≤

1

{\displaystyle 0<\alpha \leq 1}

. We also have

Lipschitz continuous ⊂ absolutely continuous ⊂ uniformly continuous ⊂ continuous.

Lipschitz continuity is named after German mathematician Rudolf Lipschitz.

Definitions

Given two metric spaces (X, dX) and (Y, dY), where dX denotes the metric on the set X and dY is the metric on set Y, a function f : X → Y is called Lipschitz continuous if there exists a real constant K ≥ 0 such that, for all x1 and x2 in X,

d

Y

(

f

(

x

1

)

,

f

(

x

2

)

)

≤

K

d

X

(

x

1

,

x

2

)

.

{\displaystyle d_{Y}(f(x_{1}),f(x_{2}))\leq Kd_{X}(x_{1},x_{2}).}

Any such K is referred to as a Lipschitz constant for the function f, and f may also be referred to as K-Lipschitz.