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Let
 be a function. We say that $f$ is of class $C^1$ if $f'$ 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). Therefore, $f\in C^\infty$ if and only if every derivative of $f$ exists.
The previous concepts can be extended to functions
 , where $f$ being of class $C^n$ amounts to asking that all the partial derivatives of order $n$ be continuous. For instance,
 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 $m$ .
Sometimes we need to talk about continuity not globally on  , but on some interval or open set.
If
 is an open set, and
 (or
 ) 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 for the multi-index notation.
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