Holder continuity

The Hölder condition (after Otto Hölder ) is a concept of mathematics , which is primarily in the theory of partial differential equations is of central importance. It is a generalization of the Lipschitz continuity .

definition

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 : ${\displaystyle U\subset \mathbb {R} }$ ${\displaystyle 0<\alpha \leq 1}$ ${\displaystyle f\colon U\rightarrow \mathbb {R} }$ ${\displaystyle \alpha }$ ${\displaystyle C}$ ${\displaystyle x,y\in U}$

${\displaystyle \vert f(x)-f(y)\vert \leq C\vert x-y\vert ^{\alpha }}$.

More generally it is called a function between two metric spaces and Hölder continuous with exponent and constant , if for all ${\displaystyle f\colon \Omega \subset E\rightarrow F}$ ${\displaystyle (E,d_{E})}$ ${\displaystyle (F,d_{F})}$ ${\displaystyle \alpha }$ ${\displaystyle C}$ ${\displaystyle x,y\in \Omega }$

${\displaystyle d_{F}(f(x),f(y))\leq C(d_{E}(x,y))^{\alpha }}$

applies.

example

For is the function with Hölder continuous to the exponent with constant , because for results in , that is . ${\displaystyle 0<\alpha \leq 1}$ ${\displaystyle f\colon {[0,\infty [}\rightarrow \mathbb {R} }$ ${\displaystyle f(x)=x^{\alpha }}$ ${\displaystyle \alpha }$ ${\displaystyle C=1}$ ${\displaystyle 0<x<y}$ ${\displaystyle 1-{\frac {x^{\alpha }}{y^{\alpha }}}\leq 1-{\frac {x}{y}}\leq {\bigg (}1-{\frac {x}{y}}{\bigg )}^{\alpha }}$ ${\displaystyle \vert x^{\alpha }-y^{\alpha }\vert \leq \vert x-y\vert ^{\alpha }}$

properties

• In the special case, the definition gives the Lipschitz continuity. In particular, every Lipschitz-continuous function is also Hölder-continuous. ${\displaystyle \alpha =1}$
• 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. ${\displaystyle (0,1]}$ ${\displaystyle \alpha =0}$ ${\displaystyle \alpha >1}$
• Every Hölder continuous function is uniformly continuous : Set for a given approximately . Then it follows as desired . ${\displaystyle \varepsilon >0}$ ${\displaystyle \delta :=(\varepsilon /C)^{1/\alpha }}$ ${\displaystyle |x-y|\leq \delta }$ ${\displaystyle |f(x)-f(y)|\leq \varepsilon }$
• 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. ${\displaystyle a\in (0,1)}$ ${\displaystyle [0,a]}$
  ${\displaystyle f(x):=\left\{{\begin{array}{cl}-1/\ln x&{\text{für }}0<x\leq a\\0&{\mbox{bei }}x=0\end{array}}\right.}$
  ${\displaystyle f}$ ${\displaystyle C>0}$ ${\displaystyle \alpha \in (0,1]}$ ${\displaystyle f(x)\leq Cx^{\alpha }}$ ${\displaystyle x\in (0,a]}$
  ${\displaystyle C\geq \lim \limits _{x\downarrow 0}{\frac {-x^{-\alpha }}{\ln x}}=\lim \limits _{x\downarrow 0}\alpha x^{-\alpha }}$
  

See also

• Holder room
• Local Holder Continuity

literature

• Hans Wilhelm Alt: Linear Functional Analysis . Springer, Berlin 2002.