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Concept
Smoothness
Summary
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called differentiability class. At the very minimum, a function could be considered smooth if it is differentiable everywhere (hence continuous). At the other end, it might also possess derivatives of all orders in its domain, in which case it is said to be infinitely differentiable and referred to as a C-infinity function (or function).
Differentiability class is a classification of functions according to the properties of their derivatives. It is a measure of the highest order of derivative that exists and is continuous for a function.
Consider an open set on the real line and a function defined on with real values. Let k be a non-negative integer. The function is said to be of differentiability class if the derivatives exist and are continuous on . If is -differentiable on , then it is at least in the class since are continuous on . The function is said to be infinitely differentiable, smooth, or of class , if it has derivatives of all orders on . (So all these derivatives are continuous functions over .) The function is said to be of class , or analytic, if is smooth (i.e., is in the class ) and its Taylor series expansion around any point in its domain converges to the function in some neighborhood of the point. is thus strictly contained in . Bump functions are examples of functions in but not in .
To put it differently, the class consists of all continuous functions. The class consists of all differentiable functions whose derivative is continuous; such functions are called continuously differentiable. Thus, a function is exactly a function whose derivative exists and is of class . In general, the classes can be defined recursively by declaring to be the set of all continuous functions, and declaring for any positive integer to be the set of all differentiable functions whose derivative is in . In particular, is contained in for every , and there are examples to show that this containment is strict ().
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