# 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**, or

*C*^{∞}**smooth**, if it has derivatives of all orders.

*f*is said to be of

**class**, or analytic, if

*C*^{ω}*f*is smooth and if it equals its Taylor series expansion around any point in its domain.

For example, the class *C*^{0} consists of all continuous functions. The class *C*^{1} consists of all differentiable functions whose derivative is continuous. In other words, a *C*^{1} function is exactly a function whose derivative exists and is of class *C*^{0}. In general, the classes *C*^{k} can be defined recursively by declaring *C*^{0} 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.

Not all differentiable functions are *C*^{1}. For example, let

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

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 **R**^{n} with values in **R**^{m}, then *f* has component functions *f*_{1}, ..., *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 exist and are continuous, where each of 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.