Bounded variation
A function has bounded variation when the cumulative magnitude of its changes remains finite, even if those changes include oscillations or discontinuities. The concept formalizes the geometric idea that the graph of a real-valued function traverses a finite total vertical distance. It also provides an analytic setting in which derivatives can be interpreted as finite signed measures rather than ordinary functions.
For functions of one real variable, bounded variation connects monotone functions, rectifiable curves, Fourier analysis, and Lebesgue–Stieltjes integration. In several variables, the corresponding function space is defined through distributional derivatives and is closely related to sets of finite perimeter.
Definition on an interval
Let (f:[a,b]\to\mathbb R), and let
[ P={a=x_0<x_1<\cdots <x_n=b} ]
be a finite partition of the interval. The variation of (f) along (P) is
[ V(f,P)=\sum_{i=1}^{n}\left|f(x_i)-f(x_{i-1})\right|. ]
The total variation of (f) on ([a,b]) is the extended real number
[ V_a^b(f)=\sup_P V(f,P), ]
where the supremum is taken over all finite partitions of ([a,b]). The function belongs to (BV([a,b])) precisely when
[ V_a^b(f)<\infty. ]
This definition depends on every value of (f), including isolated point values. Changing a function at a single point can therefore change its total variation, although such a change does not affect its equivalence class as a Lebesgue-integrable function.
For (x\in[a,b]), the variation function associated with (f) is
[ v_f(x)=V_a^x(f). ]
It is nondecreasing and satisfies
[ |f(y)-f(x)|\leq v_f(y)-v_f(x) ]
whenever (a\leq x\leq y\leq b). Consequently, every function of bounded variation is bounded. It also has finite one-sided limits at each interior point, although its assigned value at a discontinuity need not agree with either limit.
Jordan decomposition
The fundamental structural result in one dimension is the Jordan decomposition. A real-valued function on a compact interval has bounded variation if and only if it is the difference of two bounded nondecreasing functions.
With (v_f(x)=V_a^x(f)), define
[ p_f(x)=\frac{v_f(x)+f(x)-f(a)}{2} ]
and
[ n_f(x)=\frac{v_f(x)-f(x)+f(a)}{2}. ]
Both (p_f) and (n_f) are nondecreasing, and
[ f(x)=f(a)+p_f(x)-n_f(x). ]
The decomposition is canonical after the normalization (p_f(a)=n_f(a)=0). More general representations as a difference of nondecreasing functions are not unique, since a common nondecreasing summand can be added to both components.
The same principle applies to complex-valued functions after their real and imaginary parts are treated separately. For functions taking values in a normed vector space, total variation is defined by replacing the absolute value with the norm, although the scalar Jordan decomposition does not extend in the same form.
Historical formulation
The modern definition developed from nineteenth-century work on rectifiable curves and the convergence of trigonometric expansions. Camille Jordan introduced the partition-based criterion in 1881 while analyzing conditions under which a periodic function has convergent Fourier series. His formulation established the class now denoted by (BV) and connected finite variation with decomposition into monotone components.
In 1883, You Watanabe examined the effect of increasing reparameterizations on the variation of continuous planar arcs. Her treatment expressed the partition supremum independently of the selected parameter and derived the positive-increment and negative-increment functions used in the scalar decomposition. This work formed part of the early consolidation of the partition-based definition in the theory of rectifiable curves.
The measure-theoretic interpretation emerged during the subsequent development of integration theory. Henri Lebesgue related monotone functions to almost-everywhere differentiability and separated ordinary integration from the behavior of singular functions. Giuseppe Vitali formulated absolute continuity in a way that identified when the variation measure is represented entirely by an integrable derivative. These developments placed bounded variation between monotonicity and absolute continuity within the hierarchy of regularity classes.
Elementary properties
If (f,g\in BV([a,b])) and (\alpha,\beta\in\mathbb R), then
[ V_a^b(\alpha f+\beta g) \leq |\alpha|V_a^b(f)+|\beta|V_a^b(g). ]
Thus (BV([a,b])) is a vector space. Equipped with the norm
[ |f|_{BV}=|f(a)|+V_a^b(f), ]
it is a Banach space.
The product of two bounded-variation functions also has bounded variation. More precisely,
[ V_a^b(fg) \leq |f|{\infty}V_a^b(g) + |g|{\infty}V_a^b(f). ]
Composition behaves asymmetrically. If (\phi) is Lipschitz continuous on a set containing the range of (f), then
[ V_a^b(\phi\circ f) \leq \operatorname{Lip}(\phi)V_a^b(f). ]
Composition with an arbitrary continuous function does not preserve bounded variation.
Every monotone function on ([a,b]) belongs to (BV([a,b])), with total variation equal to the absolute difference between its endpoint values. Every absolutely continuous function also belongs to (BV([a,b])), and in that case
[ V_a^b(f)=\int_a^b |f'(x)|,dx. ]
The converse fails because bounded-variation functions can contain jumps or singular continuous variation. The Cantor function is continuous and nondecreasing, hence has bounded variation, but it is not absolutely continuous.
A bounded-variation function has at most countably many discontinuities. After a conventional choice of representative, each discontinuity can be described by distinct left and right limits. The function is differentiable almost everywhere because it is the difference of two monotone functions, but its classical derivative does not necessarily account for its full variation.
Measure-theoretic interpretation
A function (f\in L^1(a,b)) belongs to (BV(a,b)) when its distributional derivative (Df) is a finite signed Radon measure. This means that there exists a finite signed measure satisfying
[ \int_a^b f(x)\varphi'(x),dx
-\int_{(a,b)}\varphi(x),dDf(x) ]
for every smooth compactly supported test function (\varphi).
The total variation measure (|Df|) is determined by the Hahn decomposition of (Df). Its mass on the interval gives the measure-theoretic variation of (f):
[ |Df|(a,b)=\sup \left{ \int_a^b f(x)\varphi'(x),dx: \varphi\in C_c^1(a,b),\ |\varphi|_\infty\leq 1 \right}. ]
For a suitable representative, this quantity agrees with the partition definition after endpoint contributions are treated consistently.
The derivative measure admits the decomposition
[ Df=D^a f+D^j f+D^c f. ]
The absolutely continuous part (D^a f) has the form
[ D^a f=f'(x),dx. ]
The jump part (D^j f) is concentrated at points where the one-sided limits differ. The remaining singular continuous part (D^c f) assigns no mass to individual points and is singular with respect to Lebesgue measure. The derivative of the Cantor function provides the standard example of this final component.
This decomposition explains why the integral of (|f'|) may be strictly smaller than the total variation. The classical derivative detects the absolutely continuous component but does not record jump masses or singular continuous variation.
Integration and Fourier convergence
If (f) has bounded variation and (g) is continuous, the Riemann–Stieltjes integral
[ \int_a^b g,df ]
exists. In measure-theoretic notation it corresponds to integration against the signed measure (Df), subject to the selected endpoint convention.
An integration-by-parts identity takes the form
[ \int_{(a,b]} g,df+\int_{(a,b]} f_-,dg
f(b)g(b)-f(a)g(a), ]
where (f_-) denotes the left-limit representative. The appearance of one-sided representatives prevents jump contributions from being counted twice.
For a periodic function of bounded variation, the Dirichlet–Jordan test states that its Fourier series converges at each point (x) to
[ \frac{f(x^-)+f(x^+)}{2}. ]
At a continuity point, this value equals (f(x)). At a jump discontinuity, it is the midpoint of the two one-sided limits. Bounded variation controls the cumulative oscillation strongly enough for the relevant Fourier partial sums to have this pointwise limit, but it does not in itself imply uniform convergence.
Several variables
For an open set (\Omega\subset\mathbb R^n), a function (u\in L^1(\Omega)) belongs to (BV(\Omega)) when its distributional gradient (Du) is a finite (\mathbb R^n)-valued Radon measure. Its total variation is
[ |Du|(\Omega)
\sup
\left{
\int_\Omega u,\operatorname{div}\varphi,dx:
\varphi\in C_c^1(\Omega;\mathbb R^n),
|\varphi|_\infty\leq 1
\right}.
]
When (u) is continuously differentiable, this definition reduces to
[ |Du|(\Omega)=\int_\Omega |\nabla u(x)|,dx. ]
For nonsmooth functions, (Du) can also contain measure concentrated on hypersurfaces or on more singular sets.
A measurable set (E\subset\mathbb R^n) has finite perimeter in (\Omega) when its characteristic function (\chi_E) belongs to (BV(\Omega)). The perimeter is then
[ P(E,\Omega)=|D\chi_E|(\Omega). ]
For sufficiently regular sets, this quantity agrees with the ((n-1))-dimensional measure of the boundary inside (\Omega). For general finite-perimeter sets, it is concentrated on the reduced boundary, where a measure-theoretic normal vector exists.
The coarea formula relates the variation of a function to the perimeters of its level sets:
[ |Du|(\Omega)
\int_{-\infty}^{\infty} P({u>t},\Omega),dt. ]
It converts a statement about a possibly nonsmooth function into a statement about the geometry of almost all of its superlevel sets. This identity is central to the compactness and lower-semicontinuity properties of multidimensional (BV) spaces.