Continuity Of Set-valued Maps Research Articles

On continuity of set-valued mappings

We will show that under certain circumstances upper and lower quasi-continuity implies continuity. We also extend Fort's theorem for upper quasi-continuous mappings, by showing that every upper quasi-continuous mapping from a Baire space into compact subsets of a regular locally metrizable space is continuous at points of a dense Gδ subset of X.

Characterization of closed and robust fuzzy sets based on continuity of set-valued mappings

In the present paper, closed and robust fuzzy sets on the $n$ dimensional Euclidean space $\mathbb{R}^n$ are considered. The boundedness of their fuzzy sets are not assumed. Then, it is derived that there is a one-to-one correspondence between the class of all closed fuzzy sets and the class of all monotone decreasing left-continuous set-valued mappings, and that there is a one-to-one correspondence between the class of all robust fuzzy sets and the class of all monotone decreasing continuous set-valued mappings.

Continuity and differentiability of set-valued maps revisited in the light of tame geometry

Continuity of set-valued maps is hereby revisited: after recalling some basic concepts of variational analysis and a short description of the State-of-the-Art, we obtain as by-product two Sard type results concerning local minima of scalar and vector valued functions. Our main result though, is inscribed in the framework of tame geometry, stating that a closed-valued semialgebraic set-valued map is almost everywhere continuous (in both topological and measure-theoretic sense). The result, depending on stratification techniques, holds true in a more general setting of o-minimal (or tame) set-valued maps. Some applications are briefly discussed at the end.