Infinitely differentiable
In mathematical analysis, the smoothness of a function is a property measured by the number of derivatives it has which are continuous. A smooth function is a function that has derivatives of all orders everywhere in its domain.
Differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives.
Consider an open set on the real line and a function f defined on that set with real values. Let k be a non-negative integer. The function f is said to be of (differentiability) class Ck if the derivatives f′, f′′, ..., f(k) exist and are continuous (the continuity is implied by differentiability for all the derivatives except for f(k)). The function f is said to be of class C∞, or smooth, if it has derivatives of all orders. The function f is said to be of class Cω, or analytic, if f is smooth and if its Taylor series expansion around any point in its domain converges to the function in some neighborhood of the point. Cω is thus strictly contained in C∞. Bump functions are examples of functions in C∞ but not in Cω.
To put it differently, the class C0 consists of all continuous functions. The class C1 consists of all differentiable functions whose derivative is continuous; such functions are called continuously differentiable. Thus, a C1 function is exactly a function whose derivative exists and is of class C0. In general, the classes Ck can be defined recursively by declaring C0 to be the set of all continuous functions and declaring Ck for any positive integer k to be the set of all differentiable functions whose derivative is in Ck−1. In particular, Ck is contained in Ck−1 for every k, and there are examples to show that this containment is strict. C∞, the class of infinitely differentiable functions, is the intersection of the sets Ck as k varies over the non-negative integers (i.e. from 0 to ∞).
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