So the matter is I'm really confused about boundary conditions of magnetic fields. Prior to the boundary conditions, I thought that say you have an external magnetic field, $B_0$, now if I place a paramagnetic material in the field the resultant field inside is $B_0+\mu_0M$ and thus the field inside is greater than that of outside and that's why we are using soft iron cores may be.
But here lies the problem: if the field inside is greater than outside, now I reach to the boundary condition according to which field is continuous say there are no surface currents on the material then $$B_\text{inside} = B_\text{outside}.$$ But earlier we got that field inside is greater than outside.
My speculation (just thoughts):
Is it that the field of magnetisation also lies outside the material and is taken (included) as applied field and that's why we say that total field is continuous while magnetic field intensity $B_0/\mu_0$ is not continuous.