On Gibson functions with connected graphs

Abstract

Abstract A function f: X → Y between topological spaces is said to be a weakly Gibson function if $f(\bar G) \subseteq \overline {f(G)} $ for any open connected set G ⊆ X. We call a function f: X → Y segmentary connected if X is topological vector space and f([a, b]) is connected for every segment [a, b] ⊆ X. We show that if X is a hereditarily Baire space, Y is a metric space, f: X → Y is a Baire-one function and one of the following conditions holds: (i)X is a connected and locally connected space and f is a weakly Gibson function(ii)X is an arcwise connected space and f is a Darboux function(iii)X is a topological vector space and f is a segmentary connected function, then f has a connected graph.