Differentiability class

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In mathematical analysis, a 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.

The one-variable case[edit]

All the functions in this section will be real-valued functions of one real variable defined on some open set on the real line. Let k be a non-negative integer. A 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 the last one, f(k)). The function f is said to be of class C, or smooth, if it has derivatives of all orders. 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.

For example, the class C0 consists of all continuous functions. The class C1 consists of all differentiable functions whose derivative is continuous. In other words, 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.

The function f(x) = x2 sin(1/x) for x > 0.

Not all differentiable functions are C1. For example, let

f(x) = \begin{cases}x^2\sin{1/x}, & \text{if }x \neq 0 \\ 0, &\text{if }x = 0. \end{cases}

Applying elementary derivative rules to f shows that f is differentiable with derivative

f'(x) = \begin{cases}2x\sin{1/x} - \cos{1/x}, & \text{if }x \neq 0 \\ 0, &\text{if }x = 0.\end{cases}

Because cos 1/x oscillates as x approaches zero, f'(x) is not continuous at zero.

The higher-dimensional case[edit]

Let n and m be some positive integers. If f is a function from an open subset of Rn with values in Rm, then f has component functions f1, ..., fm. Each of these may or may not have partial derivatives. We say that f is of class Ck if all of the partial derivatives \partial^k f/\partial x_{i_1}\partial x_{i_2}\cdots\partial x_{i_k} exist and are continuous, where each of i_1, i_2, \ldots, i_k is an integer between 1 and n. The classes C and Cω are defined as before.

These criteria of differentiability can be applied to the transition functions of a differential structure. The resulting space is called a Ck manifold.

See also[edit]