Adjective
Lipschitz (not comparable)
(mathematics) (Of a real-valued real function
f
{\displaystyle f}
) Such that there exists a constant
K
{\displaystyle K}
such that whenever
x
1
{\displaystyle x_{1}}
and
x
2
{\displaystyle x_{2}}
are in the domain of
f
{\displaystyle f}
,
|
f
(
x
1
)
−
f
(
x
2
)
|
≤
K
|
x
1
−
x
2
|
{\displaystyle |f(x_{1})-f(x_{2})|\leq K|x_{1}-x_{2}|}
.
A bilipschitz function is the same thing as an injective Lipschitz function whose inverse function is also Lipschitz. Source: Internet
A function is called locally Lipschitz continuous if for every x in X there exists a neighborhood U of x such that f restricted to U is Lipschitz continuous. Source: Internet
Equivalently, if X is a locally compact metric space, then f is locally Lipschitz if and only if it is Lipschitz continuous on every compact subset of X. In spaces that are not locally compact, this is a necessary but not a sufficient condition. Source: Internet
By the result of the previous paragraph, the limit function is also Lipschitz, with the same bound for the Lipschitz constant. Source: Internet
For instance, every function that has bounded first derivatives is Lipschitz. Source: Internet
However, the discrete metric space is free in the category of bounded metric spaces and Lipschitz continuous maps, and it is free in the category of metric spaces bounded by 1 and short maps. Source: Internet