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Summary. Some new results on the continuity of K-midconvex set-valued maps are given. In particular it is shown that (under suitable assumptions on D, Y and K) a K-midconvex set-valued map is K-continuous if and only if for every linear continuous positive functional the functional , , is continuous. As a consequence of this result we obtain the equality of some set-classes connected with additive functionals and K-midconvex set-valued maps.