Höldercontinuïteit

Hölder continuity is a property of functions that describes how much a function can vary. A function f is Hölder continuous if there exist constants C, alpha, and p such that the p-norm of the difference between f(x) and f(y) is less than or equal to C times the distance between x and y raised to the power of alpha. The constant alpha is called the Hölder exponent. If alpha is between 0 and 1, the function is strictly Hölder continuous. If alpha equals 1, the function is Lipschitz continuous, which is a stronger condition. If alpha is less than 1, the function is less smooth than a Lipschitz continuous function.

The concept of Hölder continuity is important in various fields of mathematics, including analysis, differential equations, and partial differential equations. For example, it is used to establish the existence and uniqueness of solutions to certain types of equations. A function that is Hölder continuous on a compact set is uniformly continuous. However, the converse is not always true. The space of Hölder continuous functions, denoted by C^alpha(Omega), is a Banach space when endowed with the appropriate norm. This makes it a useful space for studying function properties and developing theoretical frameworks.