PlanetMath: $C^n$

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(Definition)

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

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:

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

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

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

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

exists.

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

$\begin{displaymath} \frac{\partial^2 f}{\partial x_j\partial x_i} \end{displaymath}$

exists and are all continuous for any

from

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 is of class if $\partial^\alpha f$ exist and are continuous for all multi-indices $\alpha$ with $\vert\alpha\vert\le n$ . See this page for the multi-index notation.

" is owned by drini. [ full author list (4) | owner history (1) ]

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Other names:

, $C^\infty$

Attachments:: norm (Definition) by rspuzio

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Cross-references: multi-index notation, multi-indices, open set, interval, partial derivatives, differentiable, order, derivative, continuous, class, function

This is version 10 of

, born on 2005-02-02, modified 2007-06-17.
Object id is 6701, canonical name is Cn.
Accessed 3870 times total.

Classification:

AMS MSC:	26A15 (Real functions :: Functions of one variable :: Continuity and related questions )
	26A24 (Real functions :: Functions of one variable :: Differentiation : general theory, generalized derivatives, mean-value theorems)
	26A99 (Real functions :: Functions of one variable :: Miscellaneous)
	26B05 (Real functions :: Functions of several variables :: Continuity and differentiation questions)
	46G05 (Functional analysis :: Measures, integration, derivative, holomorphy :: Derivatives)

Pending Errata and Addenda

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