Thin spherical charge layer limit ($\sigma ={\rho }_{0}a{h}^{\prime }(1)$)

For $a<r<a(1+\delta )$, a charge density ${\rho }_0{a}^2{h}^{\prime }(r/a)/(\delta r^2)$ has total electric charge $4\pi {\rho }_0{a}^3[h(1+\delta )-h(1)]/\delta$. As $\delta \to 0$, it tends to uniform surface charge on $r=a$ with surface charge density $\sigma ={\rho }_{0}a{h}^{\prime }(1)$. The limiting normal electric field has jump $\sigma /{ϵ}_0$, although every finite-thickness layer has continuous electric field at its inner edge. Taking the thin-layer limit and taking a one-sided spatial limit need not commute.