Editing Smoothness (section)

===Multivariate differentiability classes===
A function <math>f:U\subseteq\mathbb{R}^n\to\mathbb{R}</math> defined on an open set <math>U</math> of <math>\mathbb{R}^n</math> is said<ref>{{cite book|author=Henri Cartan|title=Cours de calcul différentiel|year=1977|publisher=Paris: Hermann|author-link=Henri Cartan}}</ref> to be of class <math>C^k</math> on <math>U</math>, for a positive integer <math>k</math>, if all [[partial derivatives]] 
<math display="block">\frac{\partial^\alpha f}{\partial x_1^{\alpha_1} \, \partial x_2^{\alpha_2}\,\cdots\,\partial x_n^{\alpha_n}}(y_1,y_2,\ldots,y_n)</math>
exist and are continuous, for every <math>\alpha_1,\alpha_2,\ldots,\alpha_n</math> non-negative integers, such that <math>\alpha=\alpha_1+\alpha_2+\cdots+\alpha_n\leq k</math>, and every <math>(y_1,y_2,\ldots,y_n)\in U</math>. Equivalently, <math>f</math> is of class <math>C^k</math> on <math>U</math> if the <math>k</math>-th order [[Fréchet derivative]] of <math>f</math> exists and is continuous at every point of <math>U</math>. The function <math>f</math> is said to be of class <math>C</math> or <math>C^0</math> if it is continuous on <math>U</math>. Functions of class <math>C^1</math> are also said to be ''continuously differentiable''.

A function <math>f:U\subset\mathbb{R}^n\to\mathbb{R}^m</math>, defined on an open set <math>U</math> of <math>\mathbb{R}^n</math>, is said to be of class <math>C^k</math> on <math>U</math>, for a positive integer <math>k</math>, if all of its components
<math display="block">f_i(x_1,x_2,\ldots,x_n)=(\pi_i\circ f)(x_1,x_2,\ldots,x_n)=\pi_i(f(x_1,x_2,\ldots,x_n)) \text{ for } i=1,2,3,\ldots,m</math>
are of class <math>C^k</math>, where <math>\pi_i</math> are the natural [[Projection (linear algebra)|projections]] <math>\pi_i:\mathbb{R}^m\to\mathbb{R}</math> defined by <math>\pi_i(x_1,x_2,\ldots,x_m)=x_i</math>. It is said to be of class <math>C</math> or <math>C^0</math> if it is continuous, or equivalently, if all components <math>f_i</math> are continuous, on <math>U</math>.