Continuous function between topological spaces Let $ (X,\tau_{X}) $ and $ (Y,\tau_{Y}) $ be topological spaces and $ f:X\rightarrow Y $ be a function. My question is how to show if for each $ A\subseteq Y $ , $\overline{f^{-1}(A)}$ $ \subseteq f^{-1}(\bar{A}) $  then f is continuous on X.

Hint: What does the condition tell you about $f^{-1}(F)$ for closed $F\subset Y$?
