Given: $f: X → Y, x\in X$, and $f$ is continuous at $x$, $f$ is continuous on $X$.
Prove that $f$ is continuous on $X$ if and only if for each open subset $V$ of $Y$, $f^{-1}(V)$ is an open subset of $X$.
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Your question has nothing to do with inverse functions (so I removed the tags). Here $f^{-1}(V)$ denotes the set ${x\in X\mid f(x)\in V}$ and $f^{-1}$ is not a notation for inverse function of $f$. – drhab Apr 06 '19 at 11:11
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@drhab okay thank you – souad bou chahine Apr 06 '19 at 11:14