In § {Activity: Boundary Conditions on Electric Fields}, you were asked to use an infinitesimally small Gaussian surface and an infinitesimally small Amperian loop to discover how the different components of the electric field $\EE$ behave as they cross a surface charge density. You should have chosen the surface and/or loop small enough that the field is effectively constant, except, of course, that there might be abrupt discontinuous changes where the charge resides. You should be able to show that $\EE_\parallel$ is continuous, while the discontinuity in $\EE_\perp$ is proportional to the surface charge density (and is therefore zero for continuous media), i.e. \begin{eqnarray*} \EE_{\hbox{above}}-\EE_{\hbox{below}} = \frac{\sigma}{\epsilon_0} \nn \end{eqnarray*}

In § {Activity: Gauss's Law on Cylinders and Spheres}, you will have found the electric field due to finite cylindrical and spherical shells of charge. You should check, that in the limit that these shells become infinitesimally thin, without changing the total charge on the shells, that the discontinuity in the electric field is of the form you found here. Notice, however, that your result from this activity, does not require the special symmetries of the previous activity.