In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives (differentiability class) it has over its domain.

A function of class ${\displaystyle C^{k}}$ is a function of smoothness at least k; that is, a function of class ${\displaystyle C^{k}}$ is a function that has a kth derivative that is continuous in its domain.

A function of class ${\displaystyle C^{\infty }}$ or ${\displaystyle C^{\infty }}$-function (pronounced C-infinity function) is an infinitely differentiable function, that is, a function that has derivatives of all orders (this implies that all these derivatives are continuous).

Generally, the term smooth function refers to a ${\displaystyle C^{\infty }}$-function. However, it may also mean "sufficiently differentiable" for the problem under consideration.

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 ${\displaystyle U}$ on the real line and a function ${\displaystyle f}$ defined on ${\displaystyle U}$ with real values. Let k be a non-negative integer. The function ${\displaystyle f}$ is said to be of differentiability class ${\displaystyle C^{k}}$ if the derivatives ${\displaystyle f',f'',\dots ,f^{(k)}}$ exist and are continuous on ${\displaystyle U.}$ If ${\displaystyle f}$ is ${\displaystyle k}$-differentiable on ${\displaystyle U,}$ then it is at least in the class ${\displaystyle C^{k-1}}$ since ${\displaystyle f',f'',\dots ,f^{(k-1)}}$ are continuous on ${\displaystyle U.}$ The function ${\displaystyle f}$ is said to be infinitely differentiable, smooth, or of class ${\displaystyle C^{\infty },}$ if it has derivatives of all orders on ${\displaystyle U.}$ (So all these derivatives are continuous functions over ${\displaystyle U.}$) The function ${\displaystyle f}$ is said to be of class ${\displaystyle C^{\omega },}$ or analytic, if ${\displaystyle f}$ is smooth (i.e., ${\displaystyle f}$ is in the class ${\displaystyle C^{\infty }}$) and its Taylor series expansion around any point in its domain converges to the function in some neighborhood of the point. There exist functions that are smooth but not analytic; ${\displaystyle C^{\omega }}$ is thus strictly contained in ${\displaystyle C^{\infty }.}$ Bump functions are examples of functions with this property.

To put it differently, the class ${\displaystyle C^{0}}$ consists of all continuous functions. The class ${\displaystyle C^{1}}$ consists of all differentiable functions whose derivative is continuous; such functions are called continuously differentiable. Thus, a ${\displaystyle C^{1}}$ function is exactly a function whose derivative exists and is of class ${\displaystyle C^{0}.}$ In general, the classes ${\displaystyle C^{k}}$ can be defined recursively by declaring ${\displaystyle C^{0}}$ to be the set of all continuous functions, and declaring ${\displaystyle C^{k}}$ for any positive integer ${\displaystyle k}$ to be the set of all differentiable functions whose derivative is in ${\displaystyle C^{k-1}.}$ In particular, ${\displaystyle C^{k}}$ is contained in ${\displaystyle C^{k-1}}$ for every ${\displaystyle k>0,}$ and there are examples to show that this containment is strict (${\displaystyle C^{k}\subsetneq C^{k-1}}$). The class ${\displaystyle C^{\infty }}$ of infinitely differentiable functions, is the intersection of the classes ${\displaystyle C^{k}}$ as ${\displaystyle k}$ varies over the non-negative integers.

The function $${\displaystyle f(x)={\begin{cases}x&{\mbox{if }}x\geq 0,\\0&{\text{if }}x<0\end{cases}}}$$ is continuous, but not differentiable at x = 0, so it is of class C⁰, but not of class C¹.