The main concern of this article is the stability of optimal points for sequences of functions under suitable assumptions about convergence of their epigraphs. The optimality is based on an improvement set given in the value space. Several theorems, obtained in two previous papers, about the existence and stability of optimal points related to sequences of sets in infinite dimensional spaces, are here used in the case of sequences of functions' images. Most part of the present results are new also in the case where the criteria of optimality is a cone or the spaces have finite dimension.
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