Localization of subsets of discontinuity points of a noisy function

We consider an ill-posed problem of localization of discontinuities of the first kind of a one-dimensional function, when knowing only its approximation and the error level δ in the metric of L ₂(?∞,+∞). We propose a new statement of the problem when all discontinuities are divisible into subsets, and the localization takes place for subsets of discontinuities. Assuming additionally that all discontinuities in each subset have jumps of one sign, we construct a new regular method that allows to determine the number of subsets of discontinuities, to approximate their boundaries, and to estimate the approximation accuracy.