[QUOTE]Support f:$\displaystyle R^n-> R^m$ and A (subset) $\displaystyle R^m$. Define a subset of $\displaystyle R^n$ the pre-image of the set A by$\displaystyle f"$ and denoted $\displaystyle \left(f^{-1}(A)\right)$ by

$\displaystyle \left(f^{-1}(A)\right)$= {$\displaystyle x in R^n| f(x) in A$}

Prove that $\displaystyle \left(f^{-1}(A)\right)^c$ =$\displaystyle \left(f^{-1}(A^c)\right)$ , where $\displaystyle X^c$ denotes the complement of X in$\displaystyle R^n$.

Prove the $\displaystyle f:R^n ->R^m$ is continuous <=> for all open V (subset) $\displaystyle R^m$, $\displaystyle \left(f^{-1}(V)\right)$is open in $\displaystyle R^n$

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