If $X, Y$ have the included point topology, $f: X \to Y$ is continuous iff $f$ preserves the included points
As i understand preservation we should have the following: Let $T$ be the topology in the point $p$ and $H$ in the point $q$.
Then we should have that $f(p)=q$.
($=>$). Let $V\in H=>f^{-1}(V)\in T=>p\in f^{-1}(V)=>f(p)\in V$. Thus it preserves $p$.
($<=$). Let $V\in H$. Then $q\in V=>f^{-1}(q)=p\in f^{-1}(V)$. Thus $f^{-1}(V)$ is open and $f$ is continuous.
In both situations if $V=\emptyset$ it as you wrote it above.