Editing Smoothness (section)

{{Short description|Number of derivatives of a function (mathematics)}}
{{redirect|C infinity|the extended complex plane <math>\mathbb{C}_\infty</math>|Riemann sphere}}
{{redirect|C^n|<math>\mathbb{C}^n</math>|Complex coordinate space}}
{{for|smoothness in number theory|smooth number}}
[[Image:Bump2D illustration.png|thumb|upright=1.2|A [[bump function]] is a smooth function with [[compact support]].]]

In [[mathematical analysis]], the '''smoothness''' of a [[function (mathematics)|function]] is a property measured by the number of [[Continuous function|continuous]] [[Derivative (mathematics)|derivatives]] (''differentiability class)'' it has over its [[Domain of a function|domain]].<ref>{{Cite web|url=http://mathworld.wolfram.com/SmoothFunction.html|title=Smooth Function|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-12-13|archive-date=2019-12-16|archive-url=https://web.archive.org/web/20191216043114/http://mathworld.wolfram.com/SmoothFunction.html|url-status=live}}</ref> 

A function of '''class''' <math>C^k</math> is a function of smoothness at least {{mvar|k}};  that is, a function of class <math>C^k</math> is a function that has a {{mvar|k}}th derivative that is continuous in its domain. 

A function of class <math>C^\infty</math> or <math>C^\infty</math>-function (pronounced '''C-infinity function''') is an '''infinitely   differentiable function''', that is, a function that has derivatives of all [[Order of derivation|orders]] (this implies that all these derivatives are continuous).   

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