  A function $f:X\to Y$ between topological spaces is said to be a {\it weakly
Gibson function} if $f(\overline{G})\subseteq \overline{f(G)}$ for any open
connected set \mbox{$G\subseteq X$}. We call a function $f:X\to Y$ {\it
segmentary connected} if $X$ is topological vector space and $f([a,b])$ is
connected for every segment $[a,b]\subseteq X$. We show that if $X$ is a
hereditarily Baire space, $Y$ is a metric space, \mbox{$f:X\to Y$} 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.
