# Lipschitz Condition

## Lipschitz condition

[′lip‚shits kən‚dish·ən]*b*if |ƒ(

*x*) - ƒ(

*b*)| ≤

*K*|

*x*-

*b*|, with

*K*a constant, for all

*x*in some neighborhood of

*b*.

## Lipschitz Condition

a restriction on the behavior of an increment of a function. If for any points x and *x″* in the interval *[a, b ]* the increment of a function satisfies the inequality

ǀ*f*(*x*) - *f*(*x*′) ≤ *M*ǀ *x - x′*ǀ^{α}

where 0 < α ≤ 1 and where *M* is some constant, a function *f(x)* is said to satisfy a Lipschitz condition of order α on the interval *[a, b* ]; this is written as *f(x)* ∈ Lip a. Every function satisfying a Lipschitz condition on the interval *[a, b ]* for some α > 0 is uniformly continuous on *[a, b* ]. A function having a bounded derivative on *[a, b* ] satisfies a Lipschitz condition on *[a, b ]* for any α ≤ 1.

The Lipschitz condition was first examined in 1864 by the German mathematician R. Lipschitz (1832–1903) as a sufficient condition for the convergence of the Fourier series of a function *f(x).* Although it is historically inaccurate, some mathematicians associate only the most important case of the Lipschitz condition, that of α = 1, with the name of Lipschitz; for the case α < 1 they speak of the Hölder condition.