Normal derivative of vector potential discontinuity

1. The problem statement, all variables and given/known data

In Griffiths, the following boundary condition is given without proof:

∂A_above /∂n-∂A_below /∂n=-μ₀K

for the change in the magnetic vector potential A across a surface with surface current density K, where n is the normal direction to the surface. A later problem asks for a proof of this, by using cartesian coordinates at the surface, with z perpendicular to the surface and x parallel to the current, and using the first three equations below.



2. Relevant equations

∇.A=0

A_above =A_below

B_above -B_below =μ₀(Kxn)

where n is a unit normal (I'm dropping all of the hats on my unit vectors).



The above information tell us

K=Kx

n=z



3. The attempt at a solution

First of all I have probably a really silly question because I know it's blatantly wrong, but why isn't this the case

B_above -B_below =μ₀(Kxz)

(∇xA_above )-(∇xA_below )=μ₀(Kxz)

∇x(A_above -A_below )=μ₀(Kxz)

0=μ₀(Kxz) because A_above =A_below



Aside from that, the solution states that because A_above =A_below all over the surface, ∂A/∂x and ∂A/∂y are also the same above and below the surface. Where does this come from, and why not ∂A/∂z? I can't get any further than this at present. Thanks for any help.