A differentiability class in mathematics is a class of functions which share differentiability features. The class C⁰ comprises of all continuous functions. Furthermore the class C^k comprises all functions whose derivative belongs to C^{k − 1}. The class C^∞ includes those functions which have derivatives of all orders (the smooth functions), while C^ω contains those functions which have convergent Taylor series (the analytic functions). Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... Jump to: navigation, search In mathematics, the derivative is one of the two central concepts of calculus. ... In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... In mathematics, a smooth function is one that is infinitely differentiable, i. ... Jump to: navigation, search As the degree of the Taylor series rises, it approaches the correct function. ... In mathematics, an analytic function is a function that is locally given by a convergent power series. ...

(Although called classes, the word is not meant in the modern, foundational sense; a differentiability class is not a proper class, it is an honest set.) In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ...

Note that each C^k+1 is a subset of C^k (k ≥ 0), C^∞ is a subset of C^k (k ≥ 0), and C^ω is a subset of C^∞. In the case that we consider real valued functions, these subset relations are all proper subset relations. In particular, C^ω is a proper subset of C^∞ (see an infinitely differentiable function that is not analytic). On the other hand, when we consider complex valued functions, the definition of a derivative with respect to a complex variable is much stronger, and all C^k for k≥1, C^∞, and C^ω are equal (holomorphic functions are analytic). Jump to: navigation, search A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. Every set is a subset of itself. ... A is a subset of B If X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X ⊆ Y; Y is a superset of (or includes) X; Y ⊇ X... The function Consider the real function How it is ill_behaved One can show that f has derivatives of all orders at every point on the real number line including 0. ... In complex analysis, a complex-valued function f of a complex variable is holomorphic at a point a iff it is differentiable at every point within some open disk centered at a, and is analytic at a if in some open disk centered at a it can be expanded as...

These criteria of differentiability can be applied to the transition functions of a differential structure. The resulting space will be called a C^k manifold. In topology, an atlas describes how a complicated space is glued together from simpler pieces. ... Jump to: navigation, search This page is about a higher mathematics topic. ...

See also: Hilbert's fifth problem. In mathematics, a Lie group (IPA ) is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ...