Bernstein–Doetsch-type criteria for the continuity and Lipschitz continuity of convex set-valued mappings

The Bernstein–Doetsch criterion (for convex and midconvex functionals) has been repeatedly generalized to convex and midconvex set-valued mappings F: X → 2Y; continuity and local Lipschitz continuity were understood in the sense of the Hausdorff distance. However, all such results imposed restrictive additional boundedness-type conditions on the images F(x). In this paper, the Bernstein–Doetsch criterion is generalized to arbitrary convex and midconvex set-valued mappings acting on normed linear spaces X,Y.