  The main result of this paper is a proof of the continuity of a family of
integral functionals defined on the space of functions of bounded variation
with respect to a topology under which smooth functions are dense. These
functionals occur often in the Calculus of Variations as the extension of
integral problems defined over weakly differentiable functions with linear
growth, and the result in this paper sheds light on the question of what the
'correct' extension is in this context. The result is proved via a combination
of Reshetnyak's Continuity Theorem and a map assigning a lifting
$\mu[u]\in\mathbf{M}(\Omega\times\mathbb{R}^m;\mathbb{R}^{m\times d})$ to each
$u\in BV(\Omega;\mathbb{R}^{m})$ and is valid for a large class of integrands
satisfying $|f(x,y,A)|\leq C(1+|y|^{d/(d-1)}+|A|)$. In the case where $f$
exhibits $d/(d-1)$ growth in the $y$ variable, an embedding result from the
theory of concentration-compactness is needed.
