Differentiability class

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

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 C^k 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 C^ω, 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 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^∞ is the intersection of the sets C^k as k varies over the non-negative integers. C^ω 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

Let n and m be some positive integers. If f is a function from an open subset of Rⁿ with values in R^m, then f has component functions f₁, ..., f_m. Each of these may or may not have partial derivatives. We say that f is of class C^k 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 C^ω 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 C^k manifold.

See also

• Smooth function