$f:X\to Y$ is continuous $\iff f^{-1}(A^*) \subseteq (f^{-1}(A))^*$
Suppose that $f$ is continuous, so that inverse images of open sets are open.
Let $A \subset Y$. Then $A^\ast$ is open, so $f^{-1}[A^\ast]$ is open and as $A^\ast \subseteq A$, we know that $f^{-1}[A^\ast] \subseteq f^{-1}[A]$. So as the interior of a set is the largest open subset of a set...
On the other hand, if the condition on the right is met, let $O \subseteq Y$ be open, so $O = O^\ast$. We know that $f^{-1}[O] = f^{-1}[O^\ast] \subseteq (f^{-1}[O])^\ast \subseteq f^{-1}[O]$, so ...