In the paper the description of discontinuity of the electromagnetic field in the continuum medium in terms of the extended electrodynamics in considered. The classical field theory based on the analytical Lagrange - Hamilton methods has been used as a formal apparatus. The classical theory of discontinuity waves is applied to the description of the discontinuity of the field. Dynamics of the electromagnetic field is expressed in terms of the Lagrange equations of motion in the linear approximation.
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