I was reading Purcell's E&M and the author was showing how the force on the charge distribution per unit area for a thin spherical shell with surface charge density of $\sigma$ is proportional to the average of the electric field just above and just below the surface.
He tried to show something more general to prove it. He wanted to show that the thin slab of thickness $dx$ in the diagram has force per unit area proportional to $(E_1 + E_2)/2$ where $E_1$ and $E_2$ are electric fields just to the left and right of the slab respectively. The diagram is a picture of a cross section near the surface of a charged object (like maybe the thin shell itself). Here he uses $$E_2 - E_1 = 4 \pi \sigma = 4 \pi \rho dx$$ which comes from applying Gauss's law near the slab (cgs units). My problem with this is that he is claiming that the electric field changes by $4 \pi \sigma$ as we move from left to right. But that's only if we count the electric field for the charges enclosed by the Gaussian surface. What about the charges outside the Gaussian surface? I know that their flux will be zero but that doesn't mean that the electric field due to them at $x$ and $x+dx$ will be the same.