A D18: Map $f : X \to Y$ is a

**Hölder map**from $M_X$ to $M_Y$ if and only if \begin{equation} \exists \, C, \alpha > 0 : \forall \, x, y \in X : d_Y( f(x), f(y) ) \leq C d_X(x, y)^{\alpha} \end{equation}

▾ Set of symbols

▾ Alphabet

▾ Deduction system

▾ Theory

▾ Zermelo-Fraenkel set theory

▾ Set

▾ Binary cartesian set product

▾ Binary relation

▾ Map

▾ Operation

▾ N-operation

▾ Binary operation

▾ Basic binary operation

▾ Unsigned basic binary operation

▾ Semimetric

▾ Metric

▾ Metric space

▾ Lipschitz map

▾ Alphabet

▾ Deduction system

▾ Theory

▾ Zermelo-Fraenkel set theory

▾ Set

▾ Binary cartesian set product

▾ Binary relation

▾ Map

▾ Operation

▾ N-operation

▾ Binary operation

▾ Basic binary operation

▾ Unsigned basic binary operation

▾ Semimetric

▾ Metric

▾ Metric space

▾ Lipschitz map

Formulation 0

Let $M_X = (X, d_X)$ and $M_Y = (Y, d_Y)$ each be a D1107: Metric space.

A D18: Map $f : X \to Y$ is a**Hölder map** from $M_X$ to $M_Y$ if and only if
\begin{equation}
\exists \, C, \alpha > 0 : \forall \, x, y \in X : d_Y( f(x), f(y) ) \leq C d_X(x, y)^{\alpha}
\end{equation}

A D18: Map $f : X \to Y$ is a

Also known as

Hölder continuous map

Results