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

    but I want to prove that :

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

    \{ (x,f(x)):x \in A\} <br />
is a connected set in  R^2 <br />
    The proof for this is very similar to what particlejohn suggested. Let G = \{ (x,f(x)):x \in A\} and let G=X∪Y be a partition of G into open sets X and Y. Let U = \{x\in A: (x,f(x))\in X\} and let 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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