$C^n$ Cn 2005-02-02 08:35:49 2007-06-17 23:23:06 Definition \usepackage{graphicx} %\usepackage{xypic} \usepackage{bbm} \newcommand{\Z}{\mathbbmss{Z}} \newcommand{\C}{\mathbbmss{C}} \newcommand{\R}{\mathbbmss{R}} \newcommand{\Q}{\mathbbmss{Q}} \newcommand{\mathbb}[1]{\mathbbmss{#1}} \newcommand{\figura}[1]{\begin{center}\includegraphics{#1}\end{center}} \newcommand{\figuraex}[2]{\begin{center}\includegraphics[#2]{#1}\end{center}} \newtheorem{dfn}{Definition} Let $f\colon \R\to\R$ 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 \PMlinkname{this entry}{DifferentiableFunctionsAreContinuous}). Therefore, $f\in C^\infty$ if and only if every derivative of $f$ exists. The previous concepts can be extended to functions $f\colon \R^m \to \R$, where $f$ being of class $C^n$ amounts to asking that all the partial derivatives of order $n$ be continuous. For instance, $f\colon\R^m\to \R$ 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 $m$. \subsubsection*{$C^n$ 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 $f\colon U\to \R$ (or $f\colon U\to \C$) 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 \PMlinkname{this page}{MultiIndexNotation} for the multi-index notation.