$C^n$

Let $f:ℝ\to ℝ$ 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 thefollowing relationship among these classes:

$$C^0\supset C^1\supset C^2\supset C^3\supset \mathrm{\dots }$$

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

$$C^{\mathrm{\infty }}=\bigcap _{n=0}^{\mathrm{\infty }}{C}^n.$$

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

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

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

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