HölderKontinuität

Hölder continuity is a property of continuous functions that provides a stronger notion of smoothness than ordinary continuity. A function f: X -> Y between two metric spaces (X, d_X) and (Y, d_Y) is said to be Hölder continuous with exponent alpha (where 0 < alpha <= 1) if there exists a constant C >= 0 such that for all x, y in X, the following inequality holds: d_Y(f(x), f(y)) <= C * (d_X(x, y))^alpha.

If alpha = 1, the function is called Lipschitz continuous. If alpha < 1, the function is Hölder continuous but not necessarily Lipschitz continuous. A function that is uniformly continuous is also Hölder continuous with exponent alpha approaching 0, but this is not a standard definition.

Hölder continuity is important in various fields of mathematics, including partial differential equations, functional analysis, and fractal geometry. For instance, solutions to certain partial differential equations are guaranteed to be Hölder continuous under specific conditions, which aids in their analysis and understanding. In fractal geometry, Hölder continuity is used to describe the regularity of functions defined on fractal sets. The exponent alpha quantifies the degree of smoothness, with larger values indicating smoother functions. A function with a small exponent alpha might still exhibit very irregular behavior, characteristic of fractals.