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We construct and study methods for approximating the positions (localization) of discontinuities of the first kind of a one-dimensional function. Instead of the exact function, its approximation in    and the perturbation level are known; smoothness conditions are imposed on the function outside the discontinuities. The number of discontinuities is countable, and all the discontinuities are divided into two sets: with the absolute value of the jump greater than some positive    and discontinuities satisfying a smallness condition for the value of the jump. It is required to find the number of discontinuities in the first set and localize them using the approximately given function and the perturbation level. Since the problem is ill-posed, regularization algorithms should be used for its solution. Under additional conditions on the exact function, we construct regular methods for the localization of discontinuities and obtain estimates for the accuracy of localization and for the separability threshold, which is another important characteristic of the method. The order optimality of the constructed methods on classes of functions with discontinuities is established.

New methods for the localization of discontinuities of the first kind for functions of bounded variation