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Thread: connected set

  1. #1
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    connected set

    Hi.....

    Let f:A------ R
    is a continuous function and A is a connected set
    prove that:
    { (x,f(x)):x in A}
    is a connected set
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  2. #2
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    You want to prove that $\displaystyle Z = f(A) $ is connected. Consider a continuous surjective map $\displaystyle l: A \to Z $. Let $\displaystyle Z = X \cup Y $ be a separation of $\displaystyle Z $ into 2 disjoint nonempty sets open in $\displaystyle Z $. Then $\displaystyle l^{-1}(X) $ and $\displaystyle l^{-1}(Y) $ are disjoint sets whose union is $\displaystyle A $. They are open and form a separation of $\displaystyle A $. Contradiction.
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  3. #3
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    Thank you Particlejohn,

    but I want to prove that :

    $\displaystyle \{ (x,f(x)):x \in A\}
    $ is a connected set in $\displaystyle R^2
    $
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  4. #4
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    Quote Originally Posted by karimath View Post
    but I want to prove that :

    $\displaystyle \{ (x,f(x)):x \in A\}
    $ is a connected set in $\displaystyle R^2
    $
    The proof for this is very similar to what particlejohn suggested. Let $\displaystyle G = \{ (x,f(x)):x \in A\}$ and let G=X∪Y be a partition of G into open sets X and Y. Let $\displaystyle U = \{x\in A: (x,f(x))\in X\}$ and let $\displaystyle V = \{x\in A: (x,f(x))\in Y\}$. Then U, V are open and disjoint, and their union is A ... .
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  5. #5
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    Thank you very much ,Opalg
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