Some decomposition results for functions with bounded variation are obtained by using Gagliardo's Theorem on the surjectivity of the trace operator from $W^{1;1}(\Omega)$ into $L^1(\partial \Omega)$. More precisely, we prove that every BV function can be written as the sum of a BV function without jumps and a BV function without Cantor part. Alternatively, it can be written as the sum of a BV function without jumps and a purely singular BV function (i.e., a function whose gradient is singular with respect to the Lebesgue measure). It can also be decomposed as the sum of a purely singular BV function and a BV function without Cantor part. We also prove similar results for the space BD of functions with bounded deformation. In particular, we show that every BD function can be written as the sum of a BD function without jumps and a BV function without Cantor part. Therefore, every BD function without Cantor part is the sum of a function whose symmetrized gradient belongs to $L^1$ and a BV function without Cantor part.