Smoothness

In mathematical analysis, the smoothness describes the number of times a function can be differentiated without producing discontinuities. The smoothness, or differentiability class, is an integer ${\displaystyle k}$ such that a function has all derivatives up to order ${\displaystyle k}$, and such that all of these derivatives are continuous. One says that such a function has class ${\displaystyle C^{k}}$. For example, the absolute value function ${\displaystyle f(x)=|x|}$ has class ${\displaystyle C^{0}}$, because it is continuous, but not differentiable. Generally, the term smooth function refers to a ${\displaystyle C^{\infty }}$-function, that is a function having derivatives of all orders. However, it may also mean "sufficiently differentiable" for the problem under consideration.

The usual definition is local and is therefore first made for functions defined on open subsets of Euclidean space. For functions on closed intervals, closures of open sets, or more general subsets, the same notation is also used, but its meaning depends on an additional convention, such as requiring derivatives to extend continuously to the boundary or requiring the function to be locally the restriction of a smooth function defined on an open neighborhood.

Differentiability classes are used in mathematical analysis to describe different degrees of regularity for partial differential equations. They are used in differential topology to define different classes of differentiable manifolds. For complex-valued functions, one may still speak of ${\displaystyle C^{k}}$ or ${\displaystyle C^{\infty }}$ smoothness by regarding the function as a map between real vector spaces. This should be distinguished from complex differentiability: a complex function that is complex differentiable on an open subset of ${\displaystyle \mathbb {C} }$ is holomorphic and hence analytic on that set.