A function 
satisfies the Hölder condition on two points 
and 
on an arc 
when
 |
with 
and 
positive real constants.
In some literature, functions 
satisfying the Hölder condition are sometimes said
to be (locally) 
-Hölder
continuous; moreover, 
and 
are sometimes called the Hölder exponent and Hölder constant of 
, respectively.
The Hölder condition comes up frequently in several branches of mathematics, notable among which is the study of Brownian motion in probability.
