HölderStetigkeit

Hölder continuity is a property of functions that describes how uniformly continuous a function is. A function f defined on a subset of a metric space is said to be Hölder continuous if there exist non-negative constants C and \(\alpha\), with \(\alpha > 0\), such that for all points x and y in the domain, the inequality \(d(f(x), f(y)) \leq C d(x, y)^\alpha\) holds, where \(d(\cdot, \cdot)\) denotes the metric in the respective spaces. The constant \(\alpha\) is called the Hölder exponent.

If \(\alpha = 1\), the function is Lipschitz continuous, which is a stronger condition than Hölder continuity. If \(\alpha < 1\), the function is still Hölder continuous, but not necessarily Lipschitz continuous. For \(\alpha\) approaching 0, the condition becomes weaker, and in the limit \(\alpha \to 0\), it approaches uniform continuity. However, the definition usually requires \(\alpha > 0\).

Hölder continuity is a fundamental concept in analysis, particularly in the study of partial differential equations and fractal geometry. Functions that are Hölder continuous possess a degree of smoothness that is weaker than differentiability but stronger than simple continuity. This property is crucial for establishing the existence and regularity of solutions to various mathematical problems. For instance, it plays a role in the theory of Sobolev spaces and the regularity of solutions to certain types of differential equations. The Hölder norm, derived from the Hölder continuity inequality, provides a quantitative measure of this smoothness.