Smoothness

In <a href="/facts/Mathematical_analysis/XXeR26Ih">mathematical analysis</a>, the smoothness of a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> is a property measured by the number of <a href="/facts/Continuous_function/sKbl02pB">continuous</a> <a href="/facts/Derivative_(mathematics)/7Ito70Dh">derivatives</a> (differentiability class) it has over its <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a>. 
A function of class 
 
 
 
 
 C
 
 k
 
 
 
 
 {\displaystyle C^{k}}
 
 is a function of smoothness at least k; that is, a function of class 
 
 
 
 
 C
 
 k
 
 
 
 
 {\displaystyle C^{k}}
 
 is a function that has a kth derivative that is continuous in its domain. 
A function of class 
 
 
 
 
 C
 
 ∞
 
 
 
 
 {\displaystyle C^{\infty }}
 
 or 
 
 
 
 
 C
 
 ∞
 
 
 
 
 {\displaystyle C^{\infty }}
 
-function (pronounced C-infinity function) is an infinitely differentiable function, that is, a function that has derivatives of all <a href="/facts/Order_of_derivation/7Ito70Dh">orders</a> (this implies that all these derivatives are continuous). 
Generally, the term smooth function refers to a 
 
 
 
 
 C
 
 ∞
 
 
 
 
 {\displaystyle C^{\infty }}
 
-function. However, it may also mean "sufficiently differentiable" for the problem under consideration.

Smoothness open-in-new

Smoothness