On generic continuity of maps to posets with metrics

A map f:X→Y from a Baire space X to a topological space (Y,T2) is said to be generically continuous on X if the setG(f,T2):={x∈X:f:X→(Y,T2)is continuous at x} includes a dense Gδ-set in X. In the paper, we investigate the following question: What conditions on spaces X, (Y,T1) and (Y,T2) do imply that f:X→(Y,T2) is generically continuous on X for every continuous map f:X→(Y,T1)? We answer this question for Y being a poset with a topology T2:=Td induced by a metric d on Y. It is proved that if Y is a poset with a metric d satisfying some natural conditions, then f:X→(Y,Td) is generically continuous for every Baire metric space (or Baire space) X and every continuous map f:X→(Y,↑Td), where ↑Td:={↑U:U∈Td}. Based on this result, we extend some known results for Y being hyperspaces and give an application to dynamical systems. Some related counterexamples are presented.