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 or map describes the extent to which it has derivatives that exist and vary continuously.

Given a non-negative integer ${\displaystyle k}$, a function of class ${\displaystyle C^{k}}$ is a function whose derivatives of all orders up to ${\displaystyle k}$ exist and are continuous over the function's domain.

A function of class ${\displaystyle C^{\infty }}$ is a function that is of class ${\displaystyle C^{k}}$ for every non-negative integer ${\displaystyle k}$.

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

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.