Be open and . A mapping is called Hölder continuous to the exponent if and only if there is a positive real number , so that applies to all :
.
More generally it is called a function between two metric spaces and Hölder continuous with exponent and constant , if for all
applies.
example
For is the function with Hölder continuous to the exponent with constant , because for results in , that is .
properties
In the special case, the definition gives the Lipschitz continuity. In particular, every Lipschitz-continuous function is also Hölder-continuous.
Wood exponents outside of are usually not considered. In the case of , one would get such limited but not necessarily continuous functions. In the case only constant functions fulfill the condition from the definition.
Every Hölder continuous function is uniformly continuous : Set for a given approximately . Then it follows as desired .
Not every uniformly continuous function is Holzer continuous. This is shown in the following example: Let be an arbitrarily chosen constant. The on the interval in accordance with defined function is noisy set of Heine uniformly continuous. If it were also Holzer-steady, then there would be constants and with for everyone , especially according to de l'Hospital's rule , which results in a contradiction.