Smooth map

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 C^k 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 C^ω, 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. C^ω is thus strictly contained in C^∞. Bump functions are examples of functions in C^∞ but not in C^ω.

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 C^k can be defined recursively by declaring C⁰ to be the set of all continuous functions and declaring C^k for any positive integer k to be the set of all differentiable functions whose derivative is in C^k−1. In particular, C^k is contained in C^k−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 C^k as k varies over the non-negative integers (i.e. from 0 to ∞).

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