Problem: Let $Y, E, B$ be topological spaces with $Y$ locally path connected. Suppose $p: E \rightarrow B$ is a covering map, with $g: Y \rightarrow E$ a map of sets. If $p \circ g$ is continuous, as well as $g \circ \gamma$ for every path $\gamma:[0,1] \rightarrow Y$, then $g$ is continuous.
I haven't gotten very far here. If $O$ is open in $E$, I know that $p$ is an open map, so $pO$, hence $(p \circ g)^{-1}pO= g^{-1}p^{-1}pO$, is open in $Y$. But this set only contains $g^{-1}O$.
I was thinking I would somehow need to use the lifting correspondence for covering maps.