Analogues of Transitiveness and Decomposition of Continuity

Abstract

We consider two conditions that weaken the closed graph condition and we study their properties. We show that if $X$ is a locally connected Baire space, $Y$ is a separable metrizable space and $f:X \to Y$ is a $w^*$-quasi-continuous, almost continuous and weakly Darboux function, then $f$ is continuous.