# $C^{n}$

Let $f\colon\mathbbmss{R}\to\mathbbmss{R}$ be a function. We say that $f$ is of class $C^{1}$ if $f^{\prime}$ exists and is continuous.

We also say that $f$ is of class $C^{n}$ if its $n$-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 $C^{0}$. So we get the following relationship among these classes:

 $C^{0}\supset C^{1}\supset C^{2}\supset C^{3}\supset\ldots$

Finally, the class of functions that have continuous derivatives of any order is denoted by $C^{\infty}$ and thus

 $C^{\infty}=\bigcap_{n=0}^{\infty}C^{n}.$

It holds that any function that is differentiable is also continuous (see this entry (http://planetmath.org/DifferentiableFunctionsAreContinuous)). Therefore, $f\in C^{\infty}$ if and only if every derivative of $f$ exists.

The previous concepts can be extended to functions $f\colon\mathbbmss{R}^{m}\to\mathbbmss{R}$, where $f$ being of class $C^{n}$ amounts to asking that all the partial derivatives of order $n$ be continuous. For instance, $f\colon\mathbbmss{R}^{m}\to\mathbbmss{R}$ being $C^{2}$ means that

 $\frac{\partial^{2}f}{\partial x_{j}\partial x_{i}}$

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

## $C^{n}$ functions on an open set of $\mathbbmss{R}^{m}$

Sometimes we need to talk about continuity not globally on $\mathbbmss{R}$, but on some interval or open set.

If $U\subseteq\mathbbmss{R}^{m}$ is an open set, and $f\colon U\to\mathbbmss{R}$ (or $f\colon U\to\mathbbmss{C}$) we say that $f$ is of class $C^{n}$ if $\partial^{\alpha}f$ exist and are continuous for all multi-indices $\alpha$ with $|\alpha|\leq n$. See this page (http://planetmath.org/MultiIndexNotation) for the multi-index notation.

Title $C^{n}$ Cn 2013-03-22 14:59:43 2013-03-22 14:59:43 drini (3) drini (3) 13 drini (3) Definition msc 46G05 msc 26B05 msc 26A99 msc 26A24 msc 26A15 $C^{1}$ $C^{2}$ $C^{k}$ $C^{\infty}$ Derivative SmoothFunctionsWithCompactSupport