We study the BMO-type functional
κ
ε
(
f
,
R
n
)
\kappa _{\varepsilon }(f,\mathbb {R}^n)
, which can be used to characterize bounded variation functions
f
∈
B
V
(
R
n
)
f\in \mathrm {BV}(\mathbb {R}^n)
. The
Γ
\Gamma
-limit of this functional, taken with respect to
L
l
o
c
1
L^1_{\mathrm {loc}}
-convergence, is known to be
1
4
|
D
f
|
(
R
n
)
\tfrac 14 |Df|(\mathbb {R}^n)
. We show that the
Γ
\Gamma
-limit with respect to
L
l
o
c
∞
L^{\infty }_{\mathrm {loc}}
-convergence is
\[
1
4
|
D
a
f
|
(
R
n
)
+
1
4
|
D
c
f
|
(
R
n
)
+
1
2
|
D
j
f
|
(
R
n
)
,
\tfrac 14 |D^a f|(\mathbb {R}^n)+\tfrac 14 |D^c f|(\mathbb {R}^n)+\tfrac 12 |D^j f|(\mathbb {R}^n),
\]
which agrees with the “pointwise” limit in the case of special functions of bounded varation.