Smooth Function - Differentiability Classes

Differentiability Classes

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 class Ck if the derivatives f', f'', ..., f(k) exist and are continuous (the continuity is automatic 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 it equals its Taylor series expansion around any point in its domain.

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∞ is the intersection of the sets Ck as k varies over the non-negative integers. Cω is strictly contained in C∞; for an example of this, see bump function or also below.