In this work we derive a formulation for discontinuous diffusion tensor for the Discrete Duality Finite Volume (DDFV) framework that is exact for affine solutions. In fact, DDFV methods can naturally handle anisotropic or non-linear problems on general distorded meshes. Nonetheless, a special treatment is required when the diffusion tensor is discontinuous across an internal interfaces shared by two control volumes of the mesh. In such a case, two different gradients are considered in the two subdiamonds centered at that interface and the flux conservation is imposed through an auxiliary variable at the interface.