Continuity points of quasi-continuous mappings☆☆Research supported by a Marsden fund grant, VUW 703, administered by the Royal Society of New Zealand. The first two authors are partially supported by Grant MM-701/97 of the National Fund for Scientific Research of the Bulgarian Ministry of Education, Science and Technology.

AbstractIt is known that the fragmentability of a topological space X by a metric whose topology contains the topology of X is equivalent to the existence of a winning strategy for one of the players in a special two players “fragmenting game”. In this paper we show that the absence of a winning strategy for the other player is equivalent to each of the following two properties of the space X: for every quasi-continuous mapping f:Z→X, where Z is a complete metric space, there exists a point z0∈Z at which f is continuous; for every quasi-continuous mapping f:Z→X, where Z is an α-favorable space, there exists a dense subset of Z at the points of which f is continuous.In fact, we show that the set of points of continuity of f is of the second Baire category in every non-empty open subset of Z. Using this we derive some results concerning joint continuity of separately continuous functions

On stable uniqueness in linear semi-infinite optimization

This paper is intended to provide conditions for the stability of the strong uniqueness of the optimal solution of a given linear semi-infinite optimization (LSIO) problem, in the sense of maintaining the strong uniqueness property under sufficiently small perturbations of all the data. We consider LSIO problems such that the family of gradients of all the constraints is unbounded, extending earlier results of Nürnberger for continuous LSIO problems, and of Helbig and Todorov for LSIO problems with bounded set of gradients. To do this we characterize the absolutely (affinely) stable problems, i.e., those LSIO problems whose feasible set (its affine hull, respectively) remains constant under sufficiently small perturbations.This work has been supported by MICINN of Spain, Grant MTM2008-06695-C03-01/03, by Generalitat Valenciana, by CONACyT of MX, Grant 55681, and by SECTyP-UNCuyo Res. 882/07-R