# Functional Analysis

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solution Let ${\displaystyle X\,}$ and ${\displaystyle Y\,}$ be metric spaces, and ${\displaystyle f:X\to Y\,}$ be a mapping.
(i) Prove that if ${\displaystyle f^{-1}(G)isopenwhenever[itex]G\subset Y\,}$ is open, then ${\displaystyle f\,}$ is continuous.
(ii) Prove that ${\displaystyle f\,}$ is continuous if and only if ${\displaystyle f^{-1}(F)\,}$ is closed whenever ${\displaystyle F\subset Y\,}$ is closed.