 Tangential magnetic field jump from a free surface current

With interface normal from medium 1 to medium 2, Ampère's boundary condition is $\hat{\mathbf{n}}\times ({\mathbf{H}}_2-{\mathbf{H}}_1)={\mathbf{K}}_f$. With no free surface current, tangential $H$ is continuous. Normal $B$ is continuous independently because magnetic monopoles are absent.