Smoothness

"C infinity" redirects here. For the extended complex plane ${\displaystyle \mathbb {C} _{\infty }}$, see Riemann sphere.

"C^n" redirects here. For ${\displaystyle \mathbb {C} ^{n}}$, see Complex coordinate space.

In mathematical analysis, the smoothness of a function is a property measured by the number, called differentiability class, of continuous derivatives it has over its domain.^[1]

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.