CkGlattheit

CkGlattheit, commonly written as C^k-Glattheit, is a standard regularity class in mathematics that describes the smoothness of a function. A function f defined on an open set U in Euclidean space (or between manifolds) is called C^k if its partial derivatives up to order k exist and are continuous on U. Here k is a nonnegative integer or infinity; if k = ∞, f is called C^∞ or smooth. In differential geometry, a map between manifolds is C^k when it is locally representable in coordinates by C^k functions.

Notation and scope: For a map f: X → Y, the set of C^k maps is denoted C^k(X, Y). On a compact domain K, the C^k-norm ||f||_{C^k} combines the suprema of all derivatives up to order k, turning C^k(K, Y) into a normed (and in many cases Banach) space. The hierarchy C^k ⊂ C^{k-1} shows that higher k implies stronger smoothness requirements.

Examples: Constant functions are C^k for every k. Polynomials are C^∞. The absolute value function abs(x) on R is C^0 but not C^1 at zero. The function f(x) = x|x| is C^1 but not C^2. These examples illustrate how differentiability and continuity of derivatives define the class.

Relation to other notions: C^∞ is the limit of all finite C^k classes; analytic functions form a stricter class (real analytic) than C^∞. C^k regularity is central in analysis, differential equations, and geometry, where the existence and continuity of derivatives underpin problems of approximation, stability, and curvature.

See also: smoothness, analytic functions, jets, and Sobolev spaces.