The space SBV of special functions with bounded variation, and $GSBV(\Omega)$ of generalised special functions with bounded variation, have been introduced to study the so called \emph{free discontinuity problems}. In these spaces it is possible to define a trace operator, whose definition coincides with the usual one, when $u$ is a Sobolev function. Unfortunately, due to the fact that a sequence in $(G)SBV(\Omega)$ may have jump sets getting infinitesimally close to the boundary of $\Omega$, the trace operator is not continuous. This lack of continuity leads for example to free discontinuity problems with no solution. In this talk I present a possible way to overcome this problem, by restricting our attention on a smaller class of functions. Given $\Gamma$ an $(n-1)$-dimensional set, we consider the space $(G)SBV(\Omega; \Gamma)$ of functions in $(G)SBV(\Omega)$, whose jump sets are contained in $\Gamma$. In these spaces it is possible to introduce a suitable weight function on the $\mathcal{H}^{n-1}$ measure of $\partial \Omega$, to obtain some continuity results for the trace operator. Finally, I show an application of this result, to a suitable class of free discontinuity problems with boundary conditions.