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I know we can use Gaussian pillbox method to find the discontinuity at the surface of a surface charge distribution. But can we use Gaussian pillbox method to determine electric field at a point just near the surface of a surface charge distribution?

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Assuming that there are no extra charges at this "near" region, and that it is homogeneous, I think that by applying this method you will only find that the electric field must be a continuous function in this region. As far as I know, this method is useful only for obtaining the fields at materials interfaces/discontinuities, as you have pointed.

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