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Math Help - need help

  1. #1
    Junior Member
    Joined
    Feb 2009
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    40

    need help

    consider a function f: Real \rightarrow Real. Prove that f is continuous on R iff for every subset B of Real
    f^{-1}(int B) \subseteq f^{-1} (B)
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  2. #2
    Senior Member
    Joined
    Nov 2008
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    Paris
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    354
    Hi

    This is always true.

    I think you meant: f^{-1}(intB)\subseteq int(f^{-1}(B))

    I f is continuous, then for any open set V,\ f^{-1}(V)=U is an open set. So let x be an element of f^{-1}(intB), i.e. there exists an open set V such that f(x)\in V\subseteq B. Then f^{-1}(V) is an open subset of f^{-1}(B) and contains x, therefore x\in int(f^{-1}(B)) and we can conclude: f continuous \Rightarrow f^{-1}(intB)\subseteq int(f^{-1}(B))

    Conversely, assume f ^{-1}(intB)\subseteq int(f^{-1}(B)) . Let U be an open set. Then intU=U, and we can write:

    f^{-1}(U)=f^{-1}(intU)\subseteq int(f^{-1}(U)) i.e. f^{-1}(U) \subseteq int(f^{-1}(U))

    But, for any subset A,\ A\subseteq intA means that A is open (if you never saw that, try to prove it)

    Thus for any open subset U, f^{-1}(U) is open so f is continuous, we just proved: f^{-1}(intB)\subseteq int(f^{-1}(B)) \Rightarrow f continuous.
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