====Cantor ternary function====
The [[Cantor ternary function]], also called Devil's staircase (and Cantor-Vitali functions, by some Italian authors) is the most famous example of a continuous function of bounded variation for which \eqref{e:smooth_var} fails (which was first pointed out by Vitali in {{Cite|Vi}}). In fact it is a nondecreasing function such that its derivative vanishes almost everywhere. Its generalized derivative $\mu$ vanishes on the complement of the [[Cantor set]] and the function is the prototype of [[Singular function|singular function]] in the [[Lebesgue decomposition]].
The [[Cantor ternary function]], also called Devil's staircase (and Cantor-Vitali function, by some Italian authors) is the most famous example of a continuous function of bounded variation for which \eqref{e:smooth_var} fails (which was first pointed out by Vitali in {{Cite|Vi}}). In fact it is a nondecreasing function such that its derivative vanishes almost everywhere. Its generalized derivative $\mu$ vanishes on the complement of the [[Cantor set]] and the function is the prototype of [[Singular function|singular function]] in the [[Lebesgue decomposition]].
===Historical remark===
Functions of bounded variation were introduced for the first time by C. Jordan in {{Cite|Jo}} to study the pointwise convergence of Fourier series. In particular Jordan proved the following generalization of [[Dirichlet theorem]] on the convergence of Fourier series, called [[Jordan criterion]].
Functions of bounded variation were introduced for the first time by C. Jordan in {{Cite|Jo}} to study the pointwise convergence of [[Fourier series]]. In particular Jordan proved the following generalization of the [[Dirichlet theorem]] on the convergence of Fourier series, called [[Jordan criterion]].
==Functions of several variables==
===Historical remarks===
After the introduction by Jordan of functions of bounded variations of one real variable, several authors attempted to generalize the concept to functions of more than one variable. The first attempt was made by Arzelà and Hardy in 1905, see [[Arzelà variation]] and [[Hardy variation]], followed by Vitali, Fréchet, Tonelli and Pierpont, cp. with [[Vitali variation]], [[Fréchet variation]], [[Tonelli plane variation]] and [[Pierpont variation]] (moreover, the definition of Vitali variation was also considered independently by Lebesgue and De la Vallée-Poussin). However, the point of view which became popular and it is nowadays accepted in the literature as most efficient generalization of the $1$-dimensional theory is due to De Giorgi and Fichera (see {{Cite|DG}} and {{Cite|Fi}}). Though with different definitions, the functions of bounded variation defined by De Giorgi and Fichera are equivalent (and very close in spirit) to the ''distributional theory'' described below. A promiment role in the further developing of the theory was also played by Fleming and Federer. Moreover, Krickeberg and Fleming showed, independently, that the current definition of functions of bounded variation is indeed equivalent to a slight modification of Tonelli's one {{Cite|To}}, proposed by Cesari {{Cite|Ce}}, cp. with the section '''Tonelli-Cesari variation''' below. We refer to Section 3.12 of {{Cite|AFP}} for a thorough discussion of the topic.
Functions of bouned variation in $\mathbb R^n$ can be identified with $n$-dimensional [[Current|currents]] in $\mathbb R^n$. This is the point of view of Federer, {{Cite|Fe}}, which thus derives most of the conclusions of the theory of $BV$ functions as special cases of more general theorems for normal currents,
Following Section 3.1 of {{Cite|AFP}},
The vector space of all functions of bounded variations on $\Omega$ is denoted by $BV (\Omega)$.
We assume $u\in L^1 (\Omega)$ to keep the technicalities at a minimum. However, it is possible to relax this assumption, as it is possible to define the spacel $BV_{loc} (\Omega)$ of functons of bounded local variation, i.e. such that $u\in BV (\Gamma)$ for every open $\Gamma\subset\subset\Omega$ (see {{Cite|AFP}}).
We assume $u\in L^1 (\Omega)$ to keep the technicalities at a minimum. However, it is possible to relax this assumption, as it is possible to define the space $BV_{loc} (\Omega)$ of functons of bounded local variation, i.e. such that $u\in BV (\Gamma)$ for every open $\Gamma\subset\subset\Omega$ (see {{Cite|AFP}}).
====Total variation====
* There is a measure $\mu$ such that $|D (\varphi\circ f)|\leq {\rm Lip}\, (\varphi) \mu$ for every Lipschitz function $\varphi:X\to\mathbb R$.
This definition, which found recently quite important applications, is consistent with the one-dimensional theory and with the case $X=\mathbb R^m$ given above (for the latter see the section '''Volpert chain rule''').
===Functional properties===