Thank's both, and sorry for replying late. I thing I understood the general idea of your replies but for the moment I am going to look up in the bibliography the said discontinuity at the boundary.
The first page of this short pdf from MIT sums the starting point to formulate my question:
https://ocw.mit.edu/courses/physics/8-022-physics-ii-electricity-and-magnetism-fall-2006/lecture-notes/lecture29.pdf
We can see that
∇xH = Jfree
and
∇xB =μo (Jfree +∇xM)
∇xB =μo (Jfree +JB)
And now my...