A sufficient and necessary condition of a continuous map Prove that a map $f:X\to Y$ is a continuous iff $$\text{cl}f^{-1}(A)\subseteq f^{-1}(\text{cl}A）$$ for each $A\subset Y$, where $X$ and $Y$ are both topological spaces and $\text{cl}$ denotes the closure.
In fact,for any $A\subset Y$,we have $A\subset \text{cl}A$ where $\text{cl}A$ is closed in $Y$, thus $\text{cl} f^{-1}(A)\subseteq f^{-1}(\text{cl}A）$. But how about the converse?

Let $A$ be closed in $Y$. This can be expressed by $A=\text{cl}A$. By hypothesis $f^{-1}(A)=f^{-1}(\text{cl}A)\supseteq \text{cl}f^{-1}(A)$. Can you go on from here?
