  We study the BMO-type functional $\kappa_{\varepsilon}(f,\mathbb R^n)$, which
can be used to characterize BV functions $f\in BV(\mathbb R^n)$. The
$\Gamma$-limit of this functional, taken with respect to
$L^1_{\mathrm{loc}}$-convergence, is known to be $\tfrac 14 |Df|(\mathbb R^n)$.
We show that the $\Gamma$-limit with respect to
$L^{\infty}_{\mathrm{loc}}$-convergence is \[ \tfrac 14 |D^a f|(\mathbb
R^n)+\tfrac 14 |D^c f|(\mathbb R^n)+\tfrac 12 |D^j f|(\mathbb R^n), \]
  which agrees with the ``pointwise'' limit in the case of SBV functions.
