Given metric spaces $B$ and $P$, a function $q: B \to P$ is continuous at $c \in B$ if for every $\epsilon > 0$, there exists $\delta > 0$ such that
$d_B(x, c) < \delta \implies d_P(q(x), q(c)) < \epsilon$
But if $B$ and $P$ happen to be topological spaces, $q$ is continuous if the preimage of every open subset of $P$ is open in $B$. So in this case, what would it mean for $q$ to be continuous at $c \in B$?