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For real $u\in L^1(\Omega )$, its total variation seminorm on a domain isThe vector fields have compact support inside $\Omega$, so no boundary term is charged. Taking both $\xi$ and $-\xi$ makes this equivalent to a supremum of absolute pairings. When finite, the distributional derivative $Du$ is a finite vector-valued Radon measure, and the displayed supremum is its total variation norm of a measure. For a smooth function it equals ${\int }_{\Omega }|\nabla u{|}_{2}dx$.

The function of bounded variation on a domain belongs to the Banach spaceThe L1 norm is needed because variation alone vanishes on constant functions and is only a seminorm. No smoothness of $u$ is part of this definition.

For $u\in L^1(\Omega )$, the total variation seminorm on a domain isUsing $C_c^1$ instead gives the same definition. The zero test field shows nonnegativity, and $\mathrm{TV}(0)=0$, so this is a proper extended-real function. Each test field defines a linear functional of $u$; taking their supremum proves convexity, equivalentlyIt is not strictly convex: two distinct constant functions both have zero total variation, as does every convex combination of them.

The function of bounded variation on a domain space isTotal variation is not coercive on $BV(\Omega )$: for $u_k\equiv k$ on a domain of positive measure, $\Vert u_k{\Vert }_{BV}=k|\Omega |\to \infty$, whereas $\mathrm{TV}(u_k)=0$. This is noncoercivity of total variation on constants; a mean-zero constraint can remove the constant obstruction.

For $u\in L^1(\Omega )$, its total variation seminorm on a domain isThe corresponding bounded-variation space and its zero-mean subspace areIf $u_{\Omega }=|\Omega {|}^{-1}{\int }_{\Omega }udx$, the Poincaré inequality for total variation isIn particular, $\Vert u{\Vert }_{L^1}\le C_{\Omega }\mathrm{TV}(u)$ for $u\in B{V}_0(\Omega )$.

A special function of bounded variation is a function of bounded variation on a domain whose distributional derivative has no diffuse singular part. Writing the derivative measure as an absolutely continuous part, a jump part and a remaining Cantor part,the defining condition is $D^{c}u=0$. The space is denoted $SBV(\Omega )$. Here $J_u$ is the jump set and $u^{\pm }$ are the two traces across it; ${\mathcal{L}}^n$ and ${\mathcal{H}}^{n-1}$ are Lebesgue measure and Hausdorff measure.

On an interval containing zero, $u=1_{(0,\infty )}$ has derivative a Dirac measure at zero and belongs to $SBV$: jumps are allowed. The Cantor function instead has a diffuse singular derivative and is not in $SBV$. Thus “special” separates sharp interfaces from more distributed singular variation. This makes $SBV$ useful for image segmentation and variational models with both bulk and interface energies.^[ref]