Differentiability class
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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 Ck comprises all functions whose derivative belongs to Ck − 1. The class C∞ includes those functions which have derivatives of all orders (the smooth functions), while Cω contains those functions which have convergent Taylor series (the analytic functions).
(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.)
Note that each Ck+1 is a subset of Ck (k ≥ 0), C∞ is a subset of Ck (k ≥ 0), and Cω is a subset of C∞. In the case that we consider real valued functions, these subset relations are all proper subset relations. In particular, Cω 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 Ck for k≥1, C∞, and Cω are equal (holomorphic functions are analytic).
These criteria of differentiability can be applied to the transition functions of a differential structure. The resulting space will be called a Ck manifold.
