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In this paper, we prove that the integral functional   $\mathcal {F}[u]\colon {\mathrm {BV}}(\Omega ;{\mathbb {R}}^m)\to {\mathbb {R}}$   defined by   \[\mathcal {F}[u]:=\int _\Omega f(x,u(x),\nabla u(x))\,{\mathrm {d}}x+ \int _\Omega \int _0^1 f^\infty \left ( x,u^\theta (x),\frac {{\mathrm {d}}D^s u}{{\mathrm {d}}|D^s u|}(x)\right ) {\mathrm {d}}\theta \,{\mathrm {d}}|D^su|(x)\]   is continuous over   ${\mathrm {BV}}(\Omega ;{\mathbb {R}}^m)$  , with respect to the topology of area-strict convergence, a topology in which   $({{W}}^{1,1}\cap {{C}}^\infty )(\Omega ;{\mathbb {R}}^m)$   is dense. This provides conclusive justification for the treatment of   $\mathcal {F}$   as the natural extension of the functional   \[u\mapsto \int _\Omega f(x,u(x),\nabla u(x))\,{\mathrm {d}}x,\]   defined for   $u\in {{W}}^{1,1}(\Omega ;{\mathbb {R}}^m)$  . This result is valid for a large class of integrands satisfying   $|f(x,y,A)|\leq C(1+ |y|^{d/(d-1)}+ |A|)$   and its proof makes use of Reshetnyak's Continuity Theorem combined with a lifting map   $\mu [u]\colon {\mathrm {BV}}(\Omega ;{\mathbb {R}}^m)\to {\mathbf {M}}(\Omega \times {\mathbb {R}}^m;{\mathbb {R}}^{m\times d})$  . To obtain the theorem in the case where   $f$   exhibits   $d/(d-1)$   growth in the   $y$   variable, an embedding result from the theory of concentration-compactness is also employed.

STRICTLY CONTINUOUS EXTENSION OF FUNCTIONALS WITH LINEAR GROWTH TO THE SPACE BV