An approximation of the Hammerstem integral equation with so discontinuous kernels that we are forced to look for a solution in the spaceL∞, is considered. The notion of the discrete convergence of vectors with increasing dimension to an essentially bounded function is introduced, defining a consistent system of linear piecewise integral connection operators between these spaces. Then the equation is approximated by a sequence of finite dimensional equations and the rate of the convergence of approximations is estimated.