Editing Bounded variation

{{Short description|Real function with finite total variation}}
{{Use dmy dates|date=April 2023}}
In [[mathematical analysis]], a function of '''bounded variation''', also known as '''''{{math|BV}}'' function''', is a [[real number|real]]-valued [[function (mathematics)|function]] whose [[total variation]] is bounded (finite): the [[graph of a function]] having this property is well behaved in a precise sense. For a [[continuous function]] of a single [[Variable (mathematics)|variable]], being of bounded variation means that the [[distance]] along the [[Direction (geometry, geography)|direction]] of the [[y-axis|{{math|''y''}}-axis]], neglecting the contribution of motion along [[x-axis|{{math|''x''}}-axis]], traveled by a [[point (mathematics)|point]] moving along the graph has a finite value. For a continuous function of several variables, the meaning of the definition is the same, except for the fact that the continuous path to be considered cannot be the whole graph of the given function (which is a [[Glossary of differential geometry and topology#H|hypersurface]] in this case), but can be every [[Intersection (set theory)|intersection]] of the graph itself with a [[hyperplane]] (in the case of functions of two variables, a [[Plane (mathematics)|plane]]) parallel to a fixed {{math|''x''}}-axis and to the {{math|''y''}}-axis.

Functions of bounded variation are precisely those with respect to which one may find [[Riemann–Stieltjes integral]]s of all continuous functions.

Another characterization states that the functions of bounded variation on a compact interval are exactly those {{math|''f''}} which can be written as a difference {{math|''g''&nbsp;−&nbsp;''h''}}, where both {{math|''g''}} and {{math|''h''}} are bounded [[monotonic function|monotone]]. In particular, a BV function may have discontinuities, but at most countably many.

In the case of several variables, a function {{math|''f''}} defined on an [[open subset]] {{math|Ω}} of <math>\mathbb{R}^n</math> is said to have bounded variation if its [[distribution (mathematics)|distributional derivative]] is a [[Vector-valued function|vector-valued]] finite [[Radon measure]].

One of the most important aspects of functions of bounded variation is that they form an [[Associative algebra|algebra]] of [[continuous function|discontinuous functions]] whose first derivative exists [[almost everywhere]]: due to this fact, they can and frequently are used to define [[generalized solution]]s of nonlinear problems involving [[functional (mathematics)|functional]]s, [[ordinary differential equation|ordinary]] and [[partial differential equation]]s in [[mathematics]], [[physics]] and [[engineering]].

We have the following chains of inclusions for continuous functions over a closed, bounded interval of the real line:

: '''[[Continuously differentiable]]''' ⊆ '''[[Lipschitz continuous]]''' ⊆ '''[[absolutely continuous]]''' ⊆ '''continuous and bounded variation'''  ⊆ '''[[Differentiable function|differentiable]] [[almost everywhere]]'''

==History==
According to Boris Golubov, BV functions of a single variable were first introduced by [[Camille Jordan]], in the paper {{Harv|Jordan|1881}} dealing with the convergence of [[Fourier series]]. The first successful step in the generalization of this concept to functions of several variables was due to [[Leonida Tonelli]],<ref>[[Leonida Tonelli|Tonelli]] introduced what is now called after him '''Tonelli plane variation''': for an analysis of this concept and its relations to other generalizations, see the entry "[[Total variation]]".</ref> who introduced a class of ''continuous'' BV functions in 1926 {{Harv|Cesari|1986|pp=47–48}}, to extend his [[Direct method in the calculus of variations|direct method]] for finding solutions to problems in the [[calculus of variations]] in more than one variable. Ten years after, in {{Harv|Cesari|1936}}, [[Lamberto Cesari]] ''changed the continuity requirement'' in Tonelli's definition ''to a less restrictive [[integral|integrability]] requirement'', obtaining for the first time the class of functions of bounded variation of several variables in its full generality: as Jordan did before him, he applied the concept to resolve of a problem concerning the convergence of Fourier series, but for functions of ''two variables''. After him, several authors applied BV functions to study [[Fourier series]] in several variables, [[geometric measure theory]], calculus of variations, and [[mathematical physics]]. [[Renato Caccioppoli]] and [[Ennio De Giorgi]] used them to define [[measure theory|measure]] of [[smooth function|nonsmooth]] [[boundary (topology)|boundaries]] of [[set (mathematics)|sets]] (see the entry "''[[Caccioppoli set]]''" for further information). [[Olga Arsenievna Oleinik]] introduced her view of generalized solutions for [[nonlinear partial differential equation]]s as functions from the space BV in the paper {{Harv|Oleinik|1957}}, and was able to construct a generalized solution of bounded variation of a [[First-order partial differential equation|first order]] partial differential equation in the paper {{Harv|Oleinik|1959}}: few years later, [[Edward D. Conway]] and [[Joel A. Smoller]] applied BV-functions to the study of a single [[hyperbolic equation|nonlinear hyperbolic partial differential equation]] of first order in the paper {{Harv|Conway|Smoller|1966}}, proving that the solution of the [[Cauchy problem]] for such equations is a function of bounded variation, provided the [[Cauchy boundary condition|initial value]] belongs to the same class. [[Aizik Isaakovich Vol'pert]] developed extensively a calculus for BV functions: in the paper {{Harv|Vol'pert|1967}} he proved the [[Bounded variation#Chain rule for BV functions|chain rule for BV functions]] and in the book {{Harv|Hudjaev|Vol'pert|1985}} he, jointly with his pupil [[Sergei Ivanovich Hudjaev]], explored extensively the properties of BV functions and their application. His chain rule formula was later extended by [[Luigi Ambrosio]] and [[Gianni Dal Maso]] in the paper {{Harv|Ambrosio|Dal Maso|1990}}.

==Formal definition==

=== BV functions of one variable ===
{{EquationRef|1|Definition 1.1.}} The '''[[total variation]]''' of a [[real number|real]]-valued (or more generally [[complex number|complex]]-valued) [[function (mathematics)|function]] ''f'', defined on an [[interval (mathematics)|interval]] <math>[a,b] \subset \mathbb{R}</math> is the quantity

:<math> V_a^b(f)=\sup_{P \in \mathcal{P}} \sum_{i=0}^{n_{P}-1} | f(x_{i+1})-f(x_i) |. \,</math>

where the [[supremum]] is taken over the set <math display="inline"> \mathcal{P} =\left\{P=\{ x_0, \dots , x_{n_P}\} \mid P\text{ is a partition of } [a, b]\text{ satisfying } x_i\leq x_{i+1}\text{ for } 0\leq i\leq n_P-1 \right\} </math> of all [[partition of an interval|partitions]] of the interval considered.

If ''f'' is [[derivative|differentiable]] and its derivative is Riemann-integrable, its total variation is the vertical component of the [[arc length|arc-length]] of its graph, that is to say,

:<math> V_a^b(f) = \int _a^b |f'(x)|\,\mathrm{d}x.</math>

{{EquationRef|2|Definition 1.2.}} A  real-valued function <math> f </math> on the [[real line]] is said to be of '''bounded variation''' ('''BV function''') on a chosen [[interval (mathematics)|interval]] <math>[a,b] \subset \mathbb{R}</math>  if its total variation is finite, ''i.e.''
:<math> f \in \operatorname{BV}([a,b]) \iff V_a^b(f) < +\infty </math>

It can be proved that a real function <math>f</math> is of bounded variation in <math>[a,b]</math> if and only if it can be written as the difference <math>f=f_1-f_2</math> of two non-decreasing functions <math>f_1</math> and <math>f_2</math> on <math>[a,b]</math>: this result is known as the [https://www.encyclopediaofmath.org/index.php/Jordan_decomposition_(of_a_function) Jordan decomposition of a function] and it is related to the [[Hahn decomposition theorem#Jordan measure decomposition|Jordan decomposition of a measure]].

Through the [[Stieltjes integral]], any function of bounded variation on a closed interval <math>[a, b]</math> defines a [[bounded linear functional]] on <math>C([a, b])</math>. In this special case,<ref>See for example {{harvtxt|Kolmogorov|Fomin|1969|pp=374–376}}.</ref> the [[Riesz–Markov–Kakutani representation theorem]] states that every bounded linear functional arises uniquely in this way. The normalized positive functionals or [[probability measure]]s correspond to positive non-decreasing lower [[semicontinuous function]]s. This point of view has been important in
[[spectral theory]],<ref>For a general reference on this topic, see {{harvtxt|Riesz|Szőkefalvi-Nagy|1990}}</ref> in particular in its application to [[spectral theory of ordinary differential equations|ordinary differential equations]].

===BV functions of several variables===
Functions of bounded variation, BV [[function (mathematics)|functions]], are functions whose distributional [[directional derivative|derivative]] is a [[Wikt:finite|finite]]<ref>In this context, "finite" means that its value is never [[Infinity|infinite]], i.e. it is a [[finite measure]].</ref> [[Radon measure]]. More precisely:

{{EquationRef|3|Definition 2.1.}} Let '''<math> \Omega </math>''' be an [[open subset]] of <math>\mathbb{R}^n</math>. A function '''<math> u </math>''' belonging to '''[[Lp space|<math>L^1(\Omega)</math>]]''' is said to be of '''bounded variation''' ('''BV function'''), and written

:<math> u\in \operatorname\operatorname{BV}(\Omega)</math>

if there exists a [[Finite measure|finite]] [[vector-valued function|vector]] [[Radon measure]] <math>  Du\in\mathcal M(\Omega,\mathbb{R}^n)</math> such that the following equality holds

:<math>
\int_\Omega u(x)\operatorname{div}\boldsymbol{\phi}(x)\,\mathrm{d}x = - \int_\Omega \langle\boldsymbol{\phi}, Du(x)\rangle \qquad \forall\boldsymbol{\phi}\in C_c^1(\Omega,\mathbb{R}^n)
</math>

that is, '''<math>u</math>''' defines a [[linear functional]] on the space <math>  C_c^1(\Omega,\mathbb{R}^n)</math> of [[Smooth function|continuously differentiable]] [[Vector-valued function|vector functions]] <math> \boldsymbol{\phi} </math> of [[support (mathematics)#Compact support|compact support]] contained in '''<math> \Omega </math>''': the vector [[measure (mathematics)|measure]] '''<math>Du</math>''' represents therefore the [[Distribution (mathematics)#Definitions of test functions and distributions|distributional]] or [[weak derivative|weak]] [[gradient]] of '''<math>u</math>'''.

BV can be defined equivalently in the following way.

{{EquationRef|4|Definition 2.2.}} Given a function '''<math>u</math>''' belonging to '''<math>L^1(\Omega)</math>''', the '''total variation of <math>u</math>'''<ref name="Tvar">See the entry "[[Total variation]]" for further details and more information.</ref> in <math>\Omega</math> is defined as

:<math> V(u,\Omega):=\sup\left\{\int_\Omega u(x)\operatorname{div}\boldsymbol{\phi}(x) \, \mathrm{d}x : \boldsymbol{\phi} \in C_c^1(\Omega,\mathbb{R}^n),\ \Vert\boldsymbol{\phi}\Vert_{L^\infty(\Omega)}\le 1\right\}</math>

where <math>  \Vert\;\Vert_{L^\infty(\Omega)}</math> is the [[essential supremum]] [[Norm (mathematics)|norm]]. Sometimes, especially in the theory of [[Caccioppoli set]]s, the following notation is used

:<math>\int_\Omega\vert D u\vert = V(u,\Omega)</math>

in order to emphasize that <math>V(u,\Omega)</math> is the total variation of the [[Distribution (mathematics)#Definitions of test functions and distributions|distributional]] / [[weak derivative|weak]] [[gradient]] of '''<math>u</math>'''. This notation reminds also that if '''<math>u</math>''' is of class  '''<math>C^1</math>''' (i.e. a [[continuous function|continuous]] and [[differentiable function]] having [[continuous function|continuous]] [[derivative]]s) then its [[Total variation|variation]] is exactly the [[Integral (measure theory)|integral]] of the [[absolute value]] of its [[gradient]].

The space of '''functions of bounded variation''' ('''BV functions''') can then be defined as

:<math> \operatorname\operatorname{BV}(\Omega)=\{ u\in L^1(\Omega)\colon V(u,\Omega)<+\infty\}</math>

The two definitions are equivalent since if <math>V(u,\Omega)<+\infty </math> then

:<math>\left|\int_\Omega u(x)\operatorname{div}\boldsymbol{\phi}(x) \, \mathrm{d}x \right |\leq V(u,\Omega)\Vert\boldsymbol{\phi}\Vert_{L^\infty(\Omega)}
\qquad \forall \boldsymbol{\phi}\in C_c^1(\Omega,\mathbb{R}^n)
</math>

therefore <math display="inline"> \displaystyle \boldsymbol{\phi}\mapsto\,\int_\Omega u(x)\operatorname{div}\boldsymbol{\phi}(x) \, dx</math> defines a [[continuous linear functional]] on the space <math>C_c^1(\Omega,\mathbb{R}^n)</math>. Since <math>C_c^1(\Omega,\mathbb{R}^n)
\subset C^0(\Omega,\mathbb{R}^n)</math> as a [[linear subspace]], this [[continuous linear functional]] can be extended [[continuous function|continuously]] and [[linearity|linearly]] to the whole <math>C^0(\Omega,\mathbb{R}^n)</math> by the [[Hahn–Banach theorem]]. Hence the continuous linear functional defines a [[Radon measure#Duality|Radon measure]] by the [[Riesz–Markov–Kakutani representation theorem]].

===Locally BV functions===
If the [[function space]] of [[locally integrable function]]s, i.e. [[Function (mathematics)|function]]s belonging to <math>  L^1_\text{loc}(\Omega)</math>, is considered in the preceding definitions {{EquationNote|2|1.2}}, {{EquationNote|3|2.1}} and {{EquationNote|4|2.2}} instead of the one of [[integrable function|globally integrable functions]], then the function space defined is that of '''functions of locally bounded variation'''. Precisely, developing this idea for {{EquationNote|4|definition 2.2}}, a '''[[local property|local]] variation''' is defined as follows,

: <math> V(u,U):=\sup\left\{\int_\Omega u(x)\operatorname{div}\boldsymbol{\phi}(x) \, \mathrm{d}x : \boldsymbol{\phi} \in C_c^1(U,\mathbb{R}^n),\ \Vert\boldsymbol{\phi}\Vert_{L^\infty(\Omega)}\le 1\right\}</math>

for every [[Set (mathematics)|set]] <math>  U\in\mathcal{O}_c(\Omega)</math>, having defined <math>  \mathcal{O}_c(\Omega)</math> as the set of all [[Relatively compact subspace|precompact]] [[open subset]]s of '''<math>\Omega</math>''' with respect to the standard [[topology]] of [[dimension (mathematics)|finite-dimensional]] [[vector space]]s, and correspondingly the class of functions of locally bounded variation is defined as
:<math>\operatorname{BV}_\text{loc}(\Omega)=\{ u\in L^1_\text{loc}(\Omega)\colon \, (\forall U\in\mathcal{O}_c(\Omega)) \, V(u,U)<+\infty\}</math>

===Notation===
There are basically two distinct conventions for the notation of spaces of functions of locally or globally bounded variation, and unfortunately they are quite similar: the first one, which is the one adopted in this entry, is used for example in references {{Harvtxt|Giusti|1984}} (partially), {{Harvtxt|Hudjaev|Vol'pert|1985}} (partially), {{Harvtxt|Giaquinta|Modica|Souček|1998}} and is the following one
*<math> \operatorname\operatorname{BV}(\Omega)</math> identifies the [[Space (mathematics)|space]] of functions of globally bounded variation
*<math> \operatorname\operatorname{BV}_{\text{loc}}(\Omega)</math> identifies the [[Space (mathematics)|space]] of functions of locally bounded variation
The second one, which is adopted in references {{Harvtxt|Vol'pert|1967}} and {{Harvtxt|Maz'ya|1985}} (partially), is the following:
*<math> \overline{\operatorname\operatorname{BV}}(\Omega)</math> identifies the [[Space (mathematics)|space]] of functions of globally bounded variation
*<math> \operatorname\operatorname{BV}(\Omega)</math> identifies the [[Space (mathematics)|space]] of functions of locally bounded variation

==Basic properties==
Only the properties common to [[Function (mathematics)|function]]s of one variable and to [[Function (mathematics)|function]]s of several variables will be considered in the following, and [[Mathematical proof|proof]]s will be carried on only for functions of several variables since the [[Mathematical proof|proof]] for the case of one variable is a straightforward adaptation of the several variables case: also, in each section it will be stated if the property is shared also by functions of locally bounded variation or not. References {{Harv|Giusti|1984|pp=7–9}}, {{Harv|Hudjaev|Vol'pert|1985}} and {{Harv|Màlek|Nečas|Rokyta|Růžička|1996}} are extensively used.

===BV functions have only jump-type or removable discontinuities===
In the case of one variable, the assertion is clear: for each point <math>x_0</math> in the [[interval (mathematics)|interval]] <math>[a , b]\subset\mathbb{R}</math> of definition of the function '''<math>u</math>''', either one of the following two assertions is true

:<math> \lim_{x\rightarrow x_{0^-}}\!\!\!u(x) = \!\!\!\lim_{x\rightarrow x_{0^+}}\!\!\!u(x) </math>
:<math> \lim_{x\rightarrow x_{0^-}}\!\!\!u(x) \neq \!\!\!\lim_{x\rightarrow x_{0^+}}\!\!\!u(x) </math>

while both [[Limit of a function|limits]] exist and are finite. In the case of functions of several variables, there are some premises to understand: first of all, there is a [[Linear continuum|continuum]] of [[Direction (geometry, geography)|direction]]s along which it is possible to approach a given point '''<math>x_0</math>''' belonging to the domain '''<math>\Omega</math>'''⊂<math>\mathbb{R}^n</math>. It is necessary to make precise a suitable concept of [[Limit of a function|limit]]: choosing a [[unit vector]] <math>{\boldsymbol{\hat{a}}}\in\mathbb{R}^n</math> it is possible to divide '''<math>\Omega</math>''' in two sets

:<math>\Omega_{({\boldsymbol{\hat{a}}},\boldsymbol{x}_0)} = \Omega \cap \{\boldsymbol{x}\in\mathbb{R}^n|\langle\boldsymbol{x}-\boldsymbol{x}_0,{\boldsymbol{\hat{a}}}\rangle>0\} \qquad
\Omega_{(-{\boldsymbol{\hat{a}}},\boldsymbol{x}_0)} = \Omega \cap \{\boldsymbol{x}\in\mathbb{R}^n|\langle\boldsymbol{x}-\boldsymbol{x}_0,-{\boldsymbol{\hat{a}}}\rangle>0\} </math>

Then for each point '''<math>x_0</math>''' belonging to the domain <math>\Omega\in\mathbb{R}^n</math> of the BV function '''<math>u</math>''', only one of the following two assertions is true

:<math> \lim_{\overset{\boldsymbol{x}\rightarrow \boldsymbol{x}_0}{\boldsymbol{x}\in\Omega_{({\boldsymbol{\hat{a}}},\boldsymbol{x}_0)}}}\!\!\!\!\!\!u(\boldsymbol{x}) = \!\!\!\!\!\!\!\lim_{\overset{\boldsymbol{x}\rightarrow \boldsymbol{x}_0}{\boldsymbol{x}\in\Omega_{(-{\boldsymbol{\hat{a}}},\boldsymbol{x}_0)}}}\!\!\!\!\!\!\!u(\boldsymbol{x})
</math>
:<math>  \lim_{\overset{\boldsymbol{x}\rightarrow \boldsymbol{x}_0}{\boldsymbol{x}\in\Omega_{({\boldsymbol{\hat{a}}},\boldsymbol{x}_0)}}}\!\!\!\!\!\!u(\boldsymbol{x}) \neq \!\!\!\!\!\!\!\lim_{\overset{\boldsymbol{x}\rightarrow \boldsymbol{x}_0}{\boldsymbol{x}\in\Omega_{(-{\boldsymbol{\hat{a}}},\boldsymbol{x}_0)}}}\!\!\!\!\!\!\!u(\boldsymbol{x})
</math>

or '''<math>x_0</math>''' belongs to a [[subset]] of '''<math>\Omega</math>''' having zero <math>n-1</math>-dimensional [[Hausdorff measure]]. The quantities

:<math>\lim_{\overset{\boldsymbol{x}\rightarrow \boldsymbol{x}_0}{\boldsymbol{x}\in\Omega_{({\boldsymbol{\hat{a}}},\boldsymbol{x}_0)}}}\!\!\!\!\!\!u(\boldsymbol{x})=u_{\boldsymbol{\hat a}}(\boldsymbol{x}_0) \qquad \lim_{\overset{\boldsymbol{x}\rightarrow \boldsymbol{x}_0}{\boldsymbol{x}\in\Omega_{(-{\boldsymbol{\hat{a}}},\boldsymbol{x}_0)}}}\!\!\!\!\!\!\!u(\boldsymbol{x})=u_{-\boldsymbol{\hat a}}(\boldsymbol{x}_0)</math>

are called '''approximate limits''' of the BV function '''<math>u</math>''' at the point '''<math>x_0</math>'''.

===''V''(&sdot;,&nbsp;&Omega;) is lower semi-continuous on ''L''<sup>1</sup>(&Omega;)===
The [[functional (mathematics)|functional]] <math>V(\cdot,\Omega):\operatorname\operatorname{BV}(\Omega)\rightarrow \mathbb{R}^+</math> is [[semi-continuity|lower semi-continuous]]:
to see this, choose a [[Cauchy sequence]] of BV-functions '''<math>\{u_n\}_{n\in\mathbb{N}}</math>''' converging to '''[[locally integrable function|<math>u\in L^1_\text{loc}(\Omega)</math>]]'''. Then, since all the functions of the sequence and their limit function are [[integral|integrable]] and by the definition of [[lower limit]]

:<math>\begin{align}
\liminf_{n\rightarrow\infty}V(u_n,\Omega) 
&\geq \liminf_{n\rightarrow\infty} \int_\Omega u_n(x)\operatorname{div}\, \boldsymbol{\phi}\, \mathrm{d}x \\
&\geq \int_\Omega \lim_{n\rightarrow\infty} u_n(x)\operatorname{div}\, \boldsymbol{\phi}\, \mathrm{d}x \\
&= \int_\Omega u(x)\operatorname{div}\boldsymbol{\phi}\, \mathrm{d}x \qquad\forall\boldsymbol{\phi}\in C_c^1(\Omega,\mathbb{R}^n),\quad\Vert\boldsymbol{\phi}\Vert_{L^\infty(\Omega)}\leq 1 
\end{align}</math>

Now considering the [[supremum]] on the set of functions <math>\boldsymbol{\phi}\in C_c^1(\Omega,\mathbb{R}^n)</math> such that <math>\Vert\boldsymbol{\phi}\Vert_{L^\infty(\Omega)}\leq 1 </math> then the following inequality holds true

:<math>\liminf_{n\rightarrow\infty}V(u_n,\Omega)\geq V(u,\Omega)</math>

which is exactly the definition of [[semicontinuity|lower semicontinuity]].

===BV(&Omega;) is a Banach space===
By definition '''<math>\operatorname\operatorname{BV}(\Omega)</math>''' is a [[subset]] of '''[[integrable function|<math>L^1(\Omega)</math>]]''', while [[linearity]] follows from the linearity properties of the defining [[integral]] i.e.

:<math>\begin{align}
\int_\Omega [u(x)+v(x)]\operatorname{div}\boldsymbol{\phi}(x)\,\mathrm{d}x & =
\int_\Omega u(x)\operatorname{div}\boldsymbol{\phi}(x)\,\mathrm{d}x +\int_\Omega v(x) \operatorname{div} \boldsymbol{\phi}(x)\,\mathrm{d}x = \\
& =- \int_\Omega \langle\boldsymbol{\phi}(x), Du(x)\rangle- \int_\Omega \langle \boldsymbol{\phi}(x), Dv(x)\rangle =- \int_\Omega \langle \boldsymbol{\phi}(x), [Du(x)+Dv(x)]\rangle
\end{align}
</math>

for all <math>\phi\in C_c^1(\Omega,\mathbb{R}^n)</math> therefore <math>u+v\in \operatorname\operatorname{BV}(\Omega)</math>for all <math>u,v\in \operatorname\operatorname{BV}(\Omega)</math>, and

:<math>
\int_\Omega c\cdot u(x)\operatorname{div}\boldsymbol{\phi}(x)\,\mathrm{d}x =
c \int_\Omega u(x)\operatorname{div}\boldsymbol{\phi}(x)\,\mathrm{d}x =
-c \int_\Omega \langle \boldsymbol{\phi}(x), Du(x)\rangle
</math>

for all <math> c\in\mathbb{R}</math>, therefore <math>  cu\in \operatorname\operatorname{BV}(\Omega)</math> for all <math>  u\in \operatorname\operatorname{BV}(\Omega)</math>, and all <math> c\in\mathbb{R}</math>. The proved [[vector space]] properties imply that '''<math>\operatorname\operatorname{BV}(\Omega)</math>''' is a [[vector subspace]] of '''[[Lp space|<math>L^1(\Omega)</math>]]'''. Consider now the function <math>\|\;\|_{\operatorname{BV}}:\operatorname\operatorname{BV}(\Omega)\rightarrow\mathbb{R}^+</math> defined as

:<math>\| u \|_{\operatorname{BV}} := \| u \|_{L^1} + V(u,\Omega)</math>

where <math>\| \; \|_{L^1}</math> is the usual '''[[Lp space#Lp spaces and Lebesgue integrals|<math>L^1(\Omega)</math> norm]]''': it is easy to prove that this is a [[norm (mathematics)|norm]] on '''<math>\operatorname\operatorname{BV}(\Omega)</math>'''. To see that '''<math>\operatorname\operatorname{BV}(\Omega)</math>''' is [[complete metric space|complete]] respect to it, i.e. it is a [[Banach space]], consider a [[Cauchy sequence]] <math>\{u_n\}_{n\in\mathbb{N}}</math> in '''<math>\operatorname\operatorname{BV}(\Omega)</math>'''. By definition it is also a [[Cauchy sequence]] in '''<math>L^1(\Omega)</math>''' and therefore has a [[limit of a sequence|limit]] '''<math>u</math>''' in '''<math>L^1(\Omega)</math>''': since '''<math>u_n</math>''' is bounded in '''<math>\operatorname\operatorname{BV}(\Omega)</math>''' for each '''<math>n</math>''', then <math>\Vert u \Vert_{\operatorname{BV}} < +\infty </math> by [[semicontinuity|lower semicontinuity]] of the variation <math>V(\cdot,\Omega)</math>, therefore '''<math>u</math>''' is a BV function. Finally, again by lower semicontinuity, choosing an arbitrary small positive number '''<math>\varepsilon</math>'''

:<math>\Vert u_j - u_k \Vert_{\operatorname{BV}}<\varepsilon\quad\forall j,k\geq N\in\mathbb{N} \quad\Rightarrow\quad V(u_k-u,\Omega)\leq \liminf_{j\rightarrow +\infty} V(u_k-u_j,\Omega)\leq\varepsilon</math>
From this we deduce that <math>V(\cdot,\Omega)</math> is continuous because it's a norm.

===BV(&Omega;) is not separable===
To see this, it is sufficient to consider the following example belonging to the space '''<math>\operatorname\operatorname{BV}([0,1])</math>''':<ref>The example is taken from  {{Harvtxt|Giaquinta|Modica|Souček|1998|p=331}}: see also {{harv|Kannan|Krueger|1996|loc=example 9.4.1, p. 237}}.</ref> for each 0&nbsp;<&nbsp;''&alpha;''&nbsp;<&nbsp;1 define
:<math>\chi_\alpha=\chi_{[\alpha,1]}=
\begin{cases} 0 & \mbox{if } x \notin\; [\alpha,1] \\ 
              1 & \mbox{if } x \in [\alpha,1]
\end{cases}
</math>
as the [[indicator function|characteristic function]] of the [[Interval (mathematics)#Definitions|left-closed interval]] <math>[\alpha,1]</math>. Then, choosing <math>\alpha,\beta \in [0,1]</math> such that <math>\alpha \ne \beta</math> the following relation holds true:
:<math>\Vert \chi_\alpha - \chi_\beta \Vert_{\operatorname{BV}}=2</math>
Now, in order to prove that every [[Dense set|dense subset]] of '''<math>\operatorname\operatorname{BV}(]0,1[)</math>''' cannot be [[countable set|countable]], it is sufficient to see that for every <math>\alpha\in[0,1]</math> it is possible to construct the [[Ball (mathematics)|ball]]s
:<math>B_\alpha=\left\{\psi\in \operatorname\operatorname{BV}([0,1]);\Vert \chi_\alpha - \psi \Vert_{\operatorname{BV}}\leq 1\right\}</math>
Obviously those balls are [[Disjoint sets|pairwise disjoint]], and also are an [[indexed family]] of [[set (mathematics)|set]]s whose [[index set]] is <math>[0,1]</math>. This implies that this family has the [[cardinality of the continuum]]: now, since every dense subset of <math>\operatorname\operatorname{BV}([0,1])</math> must have at least a point inside each member of this family, its cardinality is at least that of the continuum and therefore cannot a be countable subset.<ref>The same argument is used by {{Harvtxt|Kolmogorov|Fomin|1969|loc=example 7, pp. 48–49}}, in order to prove the non [[Separable space|separability]] of the space of [[bounded sequence]]s, and also {{harvtxt|Kannan|Krueger|1996|loc=example 9.4.1, p. 237}}.</ref> This example can be obviously extended to higher dimensions, and since it involves only [[Local property|local properties]], it implies that the same property is true also for '''<math>\operatorname{BV}_{loc}</math>'''.

===Chain rule for locally BV(&Omega;) functions===
[[Chain rule]]s for [[smooth function|nonsmooth function]]s are very important in [[mathematics]] and [[mathematical physics]] since there are several important [[Mathematical model|physical model]]s whose behaviors are described by [[Function (mathematics)|functions]] or [[functional (mathematics)|functional]]s with a very limited degree of [[Smooth function|smoothness]]. The following chain rule is proved in the paper {{Harv|Vol'pert|1967|p=248}}. Note all [[partial derivative]]s must be interpreted in a generalized sense, i.e., as [[Generalized derivative#Basic idea|generalized derivative]]s.

'''Theorem'''. Let <math>f:\mathbb{R}^p\rightarrow\mathbb{R}</math> be a function of class '''<math>C^1</math>''' (i.e. a [[continuous function|continuous]] and [[differentiable function]] having [[continuous function|continuous]] [[derivative]]s) and let <math>\boldsymbol{u}(\boldsymbol{x})=(u_1(\boldsymbol{x}),\ldots,u_p(\boldsymbol{x})) </math> be a function in '''<math>\operatorname\operatorname{BV}_{loc} (\Omega)</math>''' with '''<math> \Omega </math>''' being an [[open subset]] of <math> \mathbb{R}^n </math>.
Then <math>f\circ\boldsymbol{u}(\boldsymbol{x})=f(\boldsymbol{u}(\boldsymbol{x}))\in \operatorname\operatorname{BV}_{loc} (\Omega) </math> and

:<math>\frac{\partial f(\boldsymbol{u}(\boldsymbol{x}))}{\partial x_i}=\sum_{k=1}^p\frac{\partial\bar{f}(\boldsymbol{u}(\boldsymbol{x}))}{\partial u_k}\frac{\partial{u_k(\boldsymbol{x})}}{\partial x_i}
\qquad\forall i=1,\ldots,n</math>

where <math>\bar f(\boldsymbol{u}(\boldsymbol{x}))</math> is the mean value of the function at the point '''<math>x \in\Omega</math>''', defined as

:<math>\bar f(\boldsymbol{u}(\boldsymbol{x})) = \int_0^1 f\left(\boldsymbol{u}_{\boldsymbol{\hat a}}(\boldsymbol{x})t + \boldsymbol{u}_{-\boldsymbol{\hat a}}(\boldsymbol{x})(1-t)\right) \, dt</math>

A more general [[chain rule]] [[formula]] for [[lipschitz continuity|Lipschitz continuous functions]] <math>f:\mathbb{R}^p\rightarrow\mathbb{R}^s</math> has been found by [[Luigi Ambrosio]] and [[Gianni Dal Maso]] and is published in the paper {{Harv|Ambrosio|Dal Maso|1990}}. However, even this formula has very important direct consequences: we use <math>( u(\boldsymbol{x}), v(\boldsymbol{x}))</math> in place of <math>\boldsymbol u(\boldsymbol{x})</math>, where <math>v(\boldsymbol{x})</math> is also a <math>BV_{loc}</math> function. We have to assume also that <math>\bar u(\boldsymbol{x})</math> is locally integrable with respect to the measure <math>\frac{\partial v(\boldsymbol{x})}{\partial x_i}</math> for each <math>i</math>, and that <math>\bar v(\boldsymbol{x})</math> is locally integrable with respect to the measure <math>\frac{\partial u(\boldsymbol{x})}{\partial x_i}</math> for each <math>i</math>. Then choosing <math>f((u,v))=uv</math>, the preceding formula gives the '''''[[Product rule|Leibniz rule]]''''' for 'BV' functions

:<math>\frac{\partial v(\boldsymbol{x})u(\boldsymbol{x})}{\partial x_i} = {\bar u(\boldsymbol{x})}\frac{\partial v(\boldsymbol{x})}{\partial x_i} +
{\bar v(\boldsymbol{x})}\frac{\partial u(\boldsymbol{x})}{\partial x_i} </math>

==Generalizations and extensions==

=== Weighted BV functions ===
It is possible to generalize the above notion of [[total variation]] so that different variations are weighted differently. More precisely, let <math>\varphi : [0, +\infty)\longrightarrow [0, +\infty)</math> be any increasing function such that <math>\varphi(0) = \varphi(0+) =\lim_{x\rightarrow 0_+}\varphi(x) = 0</math> (the '''[[weight function]]''') and let <math>f: [0, T]\longrightarrow X </math> be a function from the [[interval (mathematics)|interval]] <math>[0 , T]</math><math>\subset \mathbb{R}</math> taking values in a [[normed vector space]] <math>X</math>. Then the <math>\boldsymbol\varphi</math>'''-variation''' of <math>f</math> over <math>[0, T]</math> is defined as

:<math>\mathop{\varphi\text{-}\operatorname{Var}}_{[0, T]} (f) := \sup \sum_{j = 0}^k \varphi \left( | f(t_{j + 1}) - f(t_j) |_X \right),</math>

where, as usual, the supremum is taken over all finite [[partition of an interval|partitions]] of the interval <math>[0, T]</math>, i.e. all the [[finite set]]s of [[real number]]s <math>t_i</math> such that

:<math>0 = t_0 < t_1 < \cdots < t_k = T.</math>

The original notion of [[Total variation|variation]] considered above is the special case of <math>\varphi</math>-variation for which the weight function is the [[identity function]]: therefore an [[integrable function]] <math>f</math> is said to be a '''weighted BV function''' (of weight <math>\varphi</math>) if and only if its <math>\varphi</math>-variation is finite.

:<math>f\in \operatorname{BV}_\varphi([0, T];X)\iff \mathop{\varphi\text{-}\operatorname{Var}}_{[0, T]} (f) <+\infty</math>

The space <math>\operatorname{BV}_\varphi([0, T];X)</math> is a [[topological vector space]] with respect to the [[norm (mathematics)|norm]]

:<math>\| f \|_{\operatorname{BV}_\varphi} := \| f \|_\infty + \mathop{\varphi\text{-}\operatorname{Var}}_{[0, T]} (f),</math>

where <math>\| f \|_{\infty}</math> denotes the usual [[supremum norm]] of ''<math>f</math>''. Weighted BV functions were introduced and studied in full generality by [[Władysław Orlicz]] and [[Julian Musielak]] in the paper {{Harvnb|Musielak|Orlicz|1959}}: [[Laurence Chisholm Young]] studied earlier the case <math>\varphi(x)=x^p</math> where ''<math>p</math>'' is a positive integer.

===SBV functions===
'''SBV functions''' ''i.e.'' ''Special functions of Bounded Variation'' were introduced by [[Luigi Ambrosio]] and [[Ennio De Giorgi]] in the paper {{Harv|Ambrosio|De Giorgi|1988}}, dealing with free discontinuity [[variational problem]]s: given an [[open subset]] '''<math> \Omega </math>''' of <math>\mathbb{R}^n</math>, the space '''<math>\operatorname{SBV}(\Omega)</math>''' is a proper [[linear subspace]] of '''<math>\operatorname\operatorname{BV}(\Omega)</math>''', since the [[weak derivative|weak]] [[gradient]] of each function belonging to it consists precisely of the [[summation|sum]] of an <math>n</math>-[[dimension]]al [[Support (mathematics)|support]] and an <math>n-1</math>-[[dimension]]al [[Support (mathematics)|support]] [[Measure (mathematics)|measure]] and ''no intermediate-dimensional terms'', as seen in the following definition.

'''Definition'''. Given a [[locally integrable function]] '''<math>u</math>''', then <math>u\in \operatorname{SBV}(\Omega) </math> if and only if

'''1.''' There exist two [[Borel function]]s <math>f</math> and <math>g</math> of [[Domain of a function|domain]] '''<math>\Omega</math>''' and [[codomain]] <math>\mathbb{R}^n</math> such that

:<math> \int_\Omega\vert f\vert \, dH^n+ \int_\Omega\vert g\vert \, dH^{n-1}<+\infty.</math>

'''2.''' For all of [[Smooth function|continuously differentiable]] [[vector-valued function|vector functions]] <math> \phi </math> of [[support (mathematics)#Compact support|compact support]] contained in '''<math> \Omega </math>''', ''i.e.'' for all <math> \phi \in
C_c^1(\Omega,\mathbb{R}^n)</math> the following formula is true:

:<math> \int_\Omega u\operatorname{div} \phi \, dH^n = \int_\Omega \langle \phi, f\rangle \, dH^n +\int_\Omega \langle \phi, g\rangle \, dH^{n-1}.</math>

where <math>H^\alpha</math> is the <math>\alpha</math>-[[dimension]]al [[Hausdorff measure]].

Details on the properties of ''SBV'' functions can be found in works cited in the bibliography section: particularly the paper {{Harv|De Giorgi|1992}} contains a useful [[bibliography]].

===BV sequences===
As particular examples of [[Banach spaces]], {{harvtxt|Dunford|Schwartz|1958|loc=Chapter IV}} consider spaces of '''sequences of bounded variation''', in addition to the spaces of functions of bounded variation. The total variation of a [[sequence (mathematics)|sequence]] ''x''&nbsp;=&nbsp;(''x''<sub>''i''</sub>) of real or complex numbers is defined by
:<math>\operatorname{TV}(x) = \sum_{i=1}^\infty |x_{i+1}-x_i|.</math>

The space of all sequences of finite total variation is denoted by BV. The norm on BV is given by
:<math>\|x\|_{\operatorname{BV}} = |x_1| + \operatorname{TV}(x) = |x_1| +  \sum_{i=1}^\infty |x_{i+1}-x_i|.</math>
With this norm, the space BV is a Banach space which is isomorphic to <math>\ell_1</math>.

The total variation itself defines a norm on a certain subspace of BV, denoted by BV<sub>0</sub>, consisting of sequences ''x'' = (''x''<sub>i</sub>) for which
:<math>\lim_{n\to\infty} x_n =0.</math>
The norm on BV<sub>0</sub> is denoted
:<math>\|x\|_{\operatorname{BV}_0} = \operatorname{TV}(x) = \sum_{i=1}^\infty |x_{i+1}-x_i|.</math>
With respect to this norm BV<sub>0</sub> becomes a Banach space as well, which is isomorphic ''and'' isometric to <math>\ell_1</math> (although not in the natural way).

===Measures of bounded variation===
A [[signed measure|signed]] (or [[complex measure|complex]]) [[Measure (mathematics)|measure]] ''<math>\mu</math>'' on a [[sigma-algebra|measurable space]] <math>(X,\Sigma)</math> is said to be of bounded variation if its [[Total variation#Total variation in measure theory|total variation]]'' <math>\Vert \mu\Vert=|\mu|(X)</math>'' is bounded: see {{harvtxt|Halmos|1950|p=123}}, {{harvtxt|Kolmogorov|Fomin|1969|p=346}} or the entry "[[Total variation]]" for further details.

==Examples==
[[File:Sin x^-1.svg|right|thumb|The function ''f''(''x'')&nbsp;=&nbsp;sin(1/''x'') is ''not'' of bounded variation on the interval <math> [0,2 / \pi] </math>.]]
As mentioned in the introduction, two large class of examples of BV functions are monotone functions, and absolutely continuous functions. For a negative example: the function

:<math>f(x) = \begin{cases} 0, & \mbox{if }x =0 \\ \sin(1/x), & \mbox{if } x \neq 0 \end{cases} </math>

is ''not'' of bounded variation on the interval <math> [0, 2/\pi]</math>

[[File:Xsin(x^-1).svg|thumb|right|The function ''f''(''x'')&nbsp;=&nbsp;''x''&nbsp;sin(1/''x'') is ''not'' of bounded variation on the interval <math> [0,2 / \pi] </math>.]]
While it is harder to see, the continuous function

:<math>f(x) = \begin{cases} 0, & \mbox{if }x =0 \\ x \sin(1/x), & \mbox{if } x \neq 0 \end{cases} </math>

is ''not'' of bounded variation on the interval <math> [0, 2/\pi]</math> either.

[[File:X^2sin(x^-1).svg|thumb|right|The function ''f''(''x'')&nbsp;=&nbsp;''x''<sup>2</sup>&nbsp;sin(1/''x'') ''is'' of bounded variation on the interval <math> [0,2 / \pi] </math>.]]
At the same time, the function

:<math>f(x) = \begin{cases} 0, & \mbox{if }x =0 \\ x^2 \sin(1/x), & \mbox{if } x \neq 0 \end{cases} </math>

is of bounded variation on the interval <math> [0,2/\pi]</math>. However, ''all three functions are of bounded variation on each interval'' <math>[a,b]</math> ''with'' <math>a>0</math>.

Every monotone, bounded function is of bounded variation. For such a function <math>f</math> on the interval <math>[a,b]</math> and any partition <math>P=\{x_0,\ldots,x_{n_P}\}</math> of this interval, it can be seen that

:<math>\sum_{i=0}^{n_P-1}|f(x_{i+1})-f(x_i)|=|f(b)-f(a)|</math>

from the fact that the sum on the left is [[Telescoping series|telescoping]]. From this, it follows that for such <math>f</math>,

:<math>V_a^b(f)=|f(b)-f(a)|.</math>

In particular, the monotone [[Cantor function]] is a well-known example of a function of bounded variation that is not [[Absolute continuity|absolutely continuous]].<ref>{{cite web | url=https://math.stackexchange.com/a/4684 | title=Real analysis - Continuous and bounded variation does not imply absolutely continuous }}</ref>

The [[Sobolev space]] '''<math> W^{1,1}(\Omega)</math>''' is a [[proper subset]] of '''<math> \operatorname\operatorname{BV}(\Omega)</math>'''. In fact, for each '''<math> u </math>''' in '''<math> W^{1,1}(\Omega) </math>''' it is possible to choose a [[Measure (mathematics)|measure]] <math> \mu:=\nabla u \mathcal L</math> (where <math> \mathcal L</math> is the [[Lebesgue measure]] on <math>\Omega</math>) such that the equality

:<math> \int u\operatorname{div}\phi = -\int \phi\, d\mu = -\int \phi \,\nabla u \qquad \forall \phi\in C_c^1 </math>

holds, since it is nothing more than the definition of [[weak derivative]], and hence holds true. One can easily find an example of a BV function which is not '''<math>W^{1,1}</math>''':  in dimension one, any step function with a non-trivial jump will do.

==Applications==

=== Mathematics ===

Functions of bounded variation have been studied in connection with the set of [[classification of discontinuities|discontinuities]] of functions and differentiability of real functions, and the following results are well-known. If <math>f</math> is a [[real number|real]] [[Function (mathematics)|function]] of bounded variation on an interval <math>[a,b]</math> then

* <math>f</math> is [[continuous function|continuous]] except at most on a [[countable set]];
* <math>f</math> has [[one-sided limit]]s everywhere (limits from the left everywhere in <math>(a,b]</math>, and from the right everywhere in <math>[a,b)</math> ;
* the [[derivative]] <math>f'(x)</math> exists [[almost everywhere]] (i.e. except for a set of [[measure zero]]).

For [[real number|real]] [[Function (mathematics)|functions]] of several real variables

* the [[Indicator function|characteristic function]] of a [[Caccioppoli set]] is a BV function: BV functions lie at the basis of the modern theory of perimeters.
* [[Minimal surface]]s are [[Graph of a function|graph]]s of BV functions: in this context, see reference {{Harv|Giusti|1984}}.

===Physics and engineering===
The ability of BV functions to deal with discontinuities has made their use widespread in the applied sciences: solutions of problems in mechanics, physics, chemical kinetics are very often representable by functions of bounded variation. The book {{Harv|Hudjaev|Vol'pert|1985}} details a very ample set of mathematical physics applications of BV functions. Also there is some modern application which deserves a brief description.

*The [[Mumford–Shah functional]]: the segmentation problem for a two-dimensional image, i.e. the problem of faithful reproduction of contours and grey scales is equivalent to the [[minimum|minimization]] of such [[Functional (mathematics)|functional]].
*[[Total variation denoising]]

==See also==
{{div col|colwidth=20em}}
* [[Renato Caccioppoli]]
* [[Caccioppoli set]]
* [[Lamberto Cesari]]
* [[Ennio De Giorgi]]
* [[Helly's selection theorem]]
* [[Locally integrable function]]
* [[Lp space|''L''<sup>''p''</sup>(&Omega;) space]]
* [[Lebesgue–Stieltjes integral]]
* [[Radon measure]]
* [[Reduced derivative]]
* [[Riemann–Stieltjes integral]]
* [[Total variation]]
* [[Quadratic variation]]
* [[p-variation]]
* [[Aizik Isaakovich Vol'pert]]
* [[Total variation denoising]]
* [[Total variation diminishing]]
{{div col end}}

==Notes==
{{Reflist|30em}}

==References==
{{Refbegin}}

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 | title =Banach Spaces and their Applications in Analysis. Proceedings of the international conference, Miami University, Oxford, OH, USA, May 22--27, 2006. In honor of Nigel Kalton's 60th birthday
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| zbl =0545.49018}}, particularly part I, chapter 1 "''Functions of bounded variation and Caccioppoli sets''". A good reference on the theory of [[Caccioppoli set]]s and their application to the [[minimal surface]] problem.
*{{Citation
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*{{Citation
| last1 = Hudjaev
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| last2 = Vol'pert
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| title = Analysis in classes of discontinuous functions and equations of mathematical physics
| place = Dordrecht–Boston–Lancaster
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| series = Mechanics: analysis
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| url = https://books.google.com/books?id=lAN0b0-1LIYC
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| isbn = 90-247-3109-7
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}}. The whole book is devoted to the theory of {{math|BV}} functions and their applications to problems in [[mathematical physics]] involving [[discontinuous function]]s and geometric objects with [[smooth function|non-smooth]] [[boundary (topology)|boundaries]].
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| series = Universitext
| pages = x+259
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}}. Maybe the most complete book reference for the theory of {{math|BV}} functions in one variable: classical results and advanced results are collected in chapter 6 "''Bounded variation''" along with several exercises. The first author was a collaborator of [[Lamberto Cesari]].
*{{Citation
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| last1 = Màlek
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| series = Applied Mathematics and Mathematical Computation
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| zbl = 0851.35002}}. One of the most complete monographs on the theory of [[Young measure]]s, strongly oriented to applications in continuum mechanics of fluids.
*{{Citation
| last = Maz'ya
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}}; particularly chapter 6, "On functions in the space {{math|BV(Ω)}}". One of the best monographs on the theory of [[Sobolev space]]s.
*{{Citation
| first = Jean Jacques
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| last1 = Musielak
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}}. In this paper, Musielak and Orlicz developed the concept of weighted {{math|BV}} functions introduced by [[Laurence Chisholm Young]] to its full generality.
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*{{Citation
| last = Vol'pert
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| language = Russian
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| year = 1967
| url = http://mi.mathnet.ru/eng/msb/v115/i2/p255
| mr = 216338 
| zbl = 0168.07402
}}. A seminal paper where [[Caccioppoli set]]s and {{math|BV}} functions are thoroughly studied and the concept of [[functional superposition]] is introduced and applied to the theory of [[partial differential equation]]s: it was also translated in English as {{Citation
| title = Spaces {{math|BV}} and quasi-linear equations
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| last1 = Vol'Pert
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| bibcode = 1967SbMat...2..225V
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| hdl-access = free
}}.

===Historical references===
*{{Citation
| last1 = Adams
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| author2-link =James A. Clarkson
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| pages = 824–854
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}}.
*{{Citation
| last1 = Alberti
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| first2 = Carlo
| title = A note on the theory of SBV functions
| journal = [[Bollettino dell'Unione Matematica Italiana]]
| series = IV Serie
| volume = 11
| issue = 2
| pages = 375–382
| year = 1997
| mr = 1459286
| zbl = 0877.49001
}}. In this paper, the authors prove the [[Compact space#Compactness of topological spaces|compactness]] of the space of SBV functions.
*{{Citation
| last1 = Ambrosio
| first1 = Luigi
| author-link = Luigi Ambrosio
| last2 = Dal Maso
| first2 = Gianni
| title = A General Chain Rule for Distributional Derivatives
| journal = [[Proceedings of the American Mathematical Society]]
| volume = 108
| issue = 3
| pages = 691
| year = 1990
| doi = 10.1090/S0002-9939-1990-0969514-3
| mr = 969514 
| zbl = 0685.49027
| doi-access = free
}}. A paper containing a very general [[chain rule]] formula for [[Function composition|composition]] of BV functions.
*{{Citation
| last1 = Ambrosio
| first1 = Luigi
| author-link = Luigi Ambrosio
| last2 = De Giorgi
| first2 = Ennio
| author2-link = Ennio De Giorgi
| title = Un nuovo tipo di funzionale del calcolo delle variazioni
|trans-title=A new kind of functional in the calculus of variations
| journal = Atti della Accademia Nazionale dei Lincei, Rendiconti della Classe di Scienze Fisiche, Matematiche e Naturali
| language =Italian
| series = VIII
| volume = LXXXII
| issue = 2
| pages = 199–210
| year = 1988
| url=http://www.bdim.eu/item?id=RLIN_1988_8_82_2_199_0
| mr = 1152641
| zbl = 0715.49014
}}. The first paper on {{math|''SBV''}} functions and related variational problems.
*{{Citation
| last = Cesari
| first = Lamberto
| author-link = Lamberto Cesari
| title = Sulle funzioni a variazione limitata 
| journal = [[Annali della Scuola Normale Superiore]]
| series = Serie II 
| volume = 5
| issue = 3–4
| pages = 299–313
| language = Italian
| year = 1936
| url = https://www.numdam.org/item?id=ASNSP_1936_2_5_3-4_299_0
| mr = 1556778
| zbl = 0014.29605
}}. Available at [http://www.numdam.org Numdam]. In the paper "''On the functions of bounded variation''" (English translation of the title) Cesari he extends the now called ''[[Total variation#Tonelli plane variation|Tonelli plane variation]]'' concept to include in the definition a subclass of the class of integrable functions.
*{{Citation
| first = Lamberto
| last = Cesari
| author-link = Lamberto Cesari
| editor-last = Montalenti
| editor-first = G.
| editor2-last = Amerio
| editor2-first = L.
| editor2-link = Luigi Amerio
| editor3-last = Acquaro
| editor3-first = G.
| editor4-last = Baiada
| editor4-first = E.
| editor5-last = Cesari
| editor5-first = L.
| editor5-link = Lamberto Cesari
| editor6-last = Ciliberto
| editor6-first = C.
| editor7-last = Cimmino
| editor7-first = G.
| editor7-link = Gianfranco Cimmino
| editor8-last = Cinquini
| editor8-first = S.
| editor9-last = De Giorgi
| editor9-first = E.
| editor9-link = Ennio De Giorgi
| editor10-last = Faedo
| editor10-first = S.
| editor10-link = Sandro Faedo
| editor11-last = Fichera
| editor11-first = G.
| editor11-link = Gaetano Fichera
| editor12-last = Galligani
| editor12-first = I.
| editor13-last = Ghizzetti
| editor13-first = A.
| editor13-link = Aldo Ghizzetti
| editor14-last = Graffi
| editor14-first = D.
| editor14-link = Dario Graffi
| editor15-last = Greco
| editor15-first = D.
| editor15-link = Donato Greco
| editor16-last = Grioli
| editor16-first = G.
| editor16-link = Giuseppe Grioli
| editor17-last = Magenes
| editor17-first = E.
| editor17-link = Enrico Magenes
| editor18-last = Martinelli
| editor18-first = E.
| editor18-link = Enzo Martinelli
| editor19-last = Pettineo
| editor19-first = B.
| editor20-last = Scorza
| editor20-first = G.
| editor20-link = Giuseppe Scorza Dragoni
| editor21-last = Vesentini
| editor21-first = E.
| editor21-link = Edoardo Vesentini
| display-editors = 4
| contribution = L'opera di Leonida Tonelli e la sua influenza nel pensiero scientifico del secolo
| title = Convegno celebrativo del centenario della nascita di Mauro Picone e Leonida Tonelli (6–9 maggio 1985)
| language = Italian
| url = http://www.lincei.it/pubblicazioni/catalogo/volume.php?lg=e&rid=32847
| series = Atti dei Convegni Lincei
| volume = 77
| year = 1986
| pages = 41–73
| place = Roma
| publisher = [[Accademia Nazionale dei Lincei]]
| archive-url = https://web.archive.org/web/20110223030014/http://www.lincei.it/pubblicazioni/catalogo/volume.php?lg=e&rid=32847
| archive-date = 23 February 2011
| url-status = dead
}}. "''The work of Leonida Tonelli and his influence on scientific thinking in this century''" (English translation of the title) is an ample commemorative article, reporting recollections of the Author about teachers and colleagues, and a detailed survey of his and theirs scientific work, presented at the ''International congress in occasion of the celebration of the centenary of birth of Mauro Picone and Leonida Tonelli'' (held in Rome on 6–9 May 1985).
*{{Citation
| last1 = Conway
| first1 = Edward D.
| last2 = Smoller
| first2 = Joel A.
| title = Global solutions of the Cauchy problem for quasi–linear first–order equations in several space variables
| journal = [[Communications on Pure and Applied Mathematics]]
| volume = 19
| issue = 1
| pages = 95–105
| year = 1966
| doi = 10.1002/cpa.3160190107
| mr = 0192161
| zbl = 0138.34701
}}. An important paper where properties of BV functions were applied to obtain a global in time [[existence theorem]] for ''single'' [[hyperbolic equation]]s of first order in any number of [[Variable (mathematics)|variables]].
*{{Citation
| first =Ennio
| last =De Giorgi
| author-link =Ennio De Giorgi
| editor-last =Amaldi
| editor-first =E.
| editor-link =Edoardo Amaldi
| editor2-last =Amerio
| editor2-first =L.
| editor2-link =Luigi Amerio
| editor3-last =Fichera
| editor3-first =G.
| editor3-link =Gaetano Fichera
| editor4-last =Gregory
| editor4-first =T.
| editor5-last =Grioli
| editor5-first =G.
| editor5-link =Giuseppe Grioli
| editor6-last =Martinelli
| editor6-first =E.
| editor6-link =Enzo Martinelli
| editor7-last =Montalenti
| editor7-first =G.
| editor8-last =Pignedoli
| editor8-first =A.
| editor8-link =Antonio Pignedoli
| editor9-last =Salvini
| editor9-link =Giorgio Salvini
| editor9-first =Giorgio
| editor10-last =Scorza Dragoni
| editor10-first =Giuseppe
| editor10-link =Giuseppe Scorza Dragoni
| contribution =Problemi variazionali con discontinuità libere <!-- |trans-title=Free-discontinuity variational problems -->
| title =Convegno internazionale in memoria di Vito Volterra (8–11 ottobre 1990)
| url =http://www.lincei.it/pubblicazioni/catalogo/volume.php?rid=32862
| language =Italian
| series =Atti dei Convegni Lincei
| volume =92
| year =1992
| pages =39–76
| place =Roma
| publisher =[[Accademia Nazionale dei Lincei]]
| issn =0391-805X
| mr =1783032
| zbl =1039.49507
| archive-url =https://web.archive.org/web/20170107005930/http://www.lincei.it/pubblicazioni/catalogo/volume.php?rid=32862
| archive-date =7 January 2017
| url-status =dead
}}. A survey paper on free-discontinuity [[calculus of variations|variational problems]] including several details on the theory of ''SBV'' functions, their applications and a rich bibliography.
*{{Citation
| last = Faleschini
| first = Bruno
| title = Sulle definizioni e proprietà delle funzioni a variazione limitata di due variabili. Nota I.
|trans-title=On the definitions and properties of functions of bounded variation of two variables. Note I
| journal = [[Bollettino dell'Unione Matematica Italiana]]
| series = Serie III
| volume = 11
| issue = 1
| pages = 80–92
| year = 1956a
| language = Italian
| url =http://www.bdim.eu/item?id=BUMI_1956_3_11_1_80_0
| mr = 80169
| zbl = 0071.27901
}}. The first part of a survey of many different definitions of "''Total variation''" and associated functions of bounded variation.
*{{Citation
| last = Faleschini
| first = Bruno
| title = Sulle definizioni e proprietà delle funzioni a variazione limitata di due variabili. Nota II.
|trans-title=On the definitions and properties of functions of bounded variation of two variables. Note I
| journal = [[Bollettino dell'Unione Matematica Italiana]]
| series = Serie III
| volume = 11
| issue = 2
| pages = 260–75
| year = 1956b
| language =Italian
| url =http://www.bdim.eu/item?id=BUMI_1956_3_11_2_260_0
| mr = 80169
| zbl = 0073.04501
}}. The second part of a survey of many different definitions of "''Total variation''" and associated functions of bounded variation.
*{{Citation
| last = Jordan
| first = Camille
| author-link = Camille Jordan
| title = Sur la série de Fourier
|trans-title=On Fourier's series
| journal = [[Comptes rendus hebdomadaires des séances de l'Académie des sciences]]
| volume = 92
| pages = 228–230
| year = 1881
| url = https://gallica.bnf.fr/ark:/12148/bpt6k7351t/f227.chemindefer
}} (at [[Gallica]]). This is, according to Boris Golubov, the first paper on functions of bounded variation.
*{{Citation
| last = Oleinik
| first = Olga A.
| author-link = Olga Arsenievna Oleinik
| title = Discontinuous solutions of non-linear differential equations
| journal = [[Uspekhi Matematicheskikh Nauk]]
| volume = 12
| issue = 3(75)
| pages = 3–73
| year = 1957
| url = http://mi.mathnet.ru/eng/umn/v12/i3/p3
| zbl = 0080.07701
}} ({{in lang|ru}}). An important paper where the author describes generalized solutions of [[nonlinear equation|nonlinear]] [[partial differential equation]]s as {{math|BV}} functions.
*{{Citation
| last = Oleinik
| first = Olga A.
| author-link = Olga Arsenievna Oleinik
| title = Construction of a generalized solution of the Cauchy problem for a quasi-linear equation of first order by the introduction of "vanishing viscosity"
| journal = [[Uspekhi Matematicheskikh Nauk]]
| volume = 14
| issue = 2(86)
| pages = 159–164
| year = 1959
| url = http://mi.mathnet.ru/eng/umn/v14/i2/p159
| zbl = 0096.06603
}} ({{in lang|ru}}). An important paper where the author constructs a [[weak solution]] in BV for a [[nonlinear equation|nonlinear]] [[partial differential equation]] with the method of [[vanishing viscosity]].
*[[Tony F. Chan]] and [https://sites.google.com/view/jackieshen/ Jianhong (Jackie) Shen] (2005), [https://web.archive.org/web/20080117220948/http://jackieneoshen.googlepages.com/ImagingNewEra.html ''Image Processing and Analysis - Variational, PDE, Wavelet, and Stochastic Methods''], SIAM Publisher, {{ISBN|0-89871-589-X}} (with in-depth coverage and extensive applications of Bounded Variations in modern image processing, as started by Rudin, Osher, and Fatemi).
{{Refend}}

==External links==

=== Theory ===
* {{springer
| title= Variation of a function
| id= V/v096110
| last= Golubov
| first= Boris I.
| last2= Vitushkin
| first2= Anatolii G.
| author2-link= Anatolii Georgievich Vitushkin
}}
*{{planetmath reference|urlname=BVFunction|title=BV function}}.
*{{MathWorld
|author=Rowland, Todd
|author2=Weisstein, Eric W.
|name-list-style=amp
|title=Bounded Variation
|urlname=BoundedVariation}}
*[https://www.encyclopediaofmath.org/index.php/Function_of_bounded_variation Function of bounded variation] at [https://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]

===Other===
* Luigi Ambrosio [https://cvgmt.sns.it/people/ambrosio/ home page] at the [[Scuola Normale Superiore di Pisa]]. Academic home page (with preprints and publications) of one of the contributors to the theory and applications of BV functions.
* [https://cvgmt.sns.it/ Research Group in Calculus of Variations and Geometric Measure Theory], [[Scuola Normale Superiore di Pisa]].

{{Functional analysis}}

{{PlanetMath attribution|id=6969|title=BV function}}

{{DEFAULTSORT:Bounded Variation}}
[[Category:Real analysis]]
[[Category:Calculus of variations]]
[[Category:Measure theory]]