We also say that is of class if its -th derivative exists and is continuous (and therefore all other previous derivatives exist and are continuous too).
The class of continuous functions is denoted by . So we get the following relationship among these classes:
Finally, the class of functions that have continuous derivatives of any order is denoted by and thus
It holds that any function that is differentiable is also continuous (see this entry (http://planetmath.org/DifferentiableFunctionsAreContinuous)). Therefore, if and only if every derivative of exists.
exists and are all continuous for any from to .
functions on an open set of
If is an open set, and (or ) we say that is of class if exist and are continuous for all multi-indices with . See this page (http://planetmath.org/MultiIndexNotation) for the multi-index notation.
|Date of creation||2013-03-22 14:59:43|
|Last modified on||2013-03-22 14:59:43|
|Last modified by||drini (3)|