Let f: be a function. We say that f is of class C1 if f exists and is continuousMathworldPlanetmathPlanetmath.

We also say that f is of class Cn if its n-th derivativeMathworldPlanetmathPlanetmath exists and is continuous (and therefore all other previous derivatives exist and are continuous too).

The class of continuous functions is denoted by C0. So we get the following relationship among these classes:


Finally, the class of functions that have continuous derivatives of any order is denoted by C and thus


It holds that any function that is differentiableMathworldPlanetmathPlanetmath is also continuous (see this entry ( Therefore, fC if and only if every derivative of f exists.

The previous concepts can be extended to functions f:m, where f being of class Cn amounts to asking that all the partial derivativesMathworldPlanetmath of order n be continuous. For instance, f:m being C2 means that


exists and are all continuous for any i,j from 1 to m.

Cn functions on an open set of m

Sometimes we need to talk about continuity not globally on , but on some interval or open set.

If Um is an open set, and f:U (or f:U) we say that f is of class Cn if αf exist and are continuous for all multi-indices α with |α|n. See this page ( for the multi-index notation.

Title Cn
Canonical name Cn
Date of creation 2013-03-22 14:59:43
Last modified on 2013-03-22 14:59:43
Owner drini (3)
Last modified by drini (3)
Numerical id 13
Author drini (3)
Entry type Definition
Classification msc 46G05
Classification msc 26B05
Classification msc 26A99
Classification msc 26A24
Classification msc 26A15
Synonym C1
Synonym C2
Synonym Ck
Synonym C
Related topic Derivative
Related topic SmoothFunctionsWithCompactSupport