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  1. #1
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    Prove that D is connected

    Let D \subset R, and let f: D-->R be continuous. Prove that D is connected if { x,f(x): x \in D}, the graph of f, is a connected subset of R^2
    Last edited by wopashui; November 25th 2011 at 06:03 PM.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Re: Prove that D is connected

    Quote Originally Posted by wopashui View Post
    Let D \subset R, and let f: D-->R be continuous. Prove that D is connected if { x,f(x): x \in D}, the graph of f, is a connected subset of R^2
    This is an if and only if. Let \Gamma_f denote the graph, note then that D=\pi_1(\Gamma_f) where \pi_1:R^2\to R is the canonical projection onto the first coordinate--why does this tell us D's connected. Conversely, if D is connected then \Gamma_f=h(D) where h: D\to R^2: x \mapsto (x,f(x))--why is h continuous and why does this tell us that \Gamma_f is connected?
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  3. #3
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    Re: Prove that D is connected

    Quote Originally Posted by Drexel28 View Post
    This is an if and only if. Let \Gamma_f denote the graph, note then that D=\pi_1(\Gamma_f) where \pi_1:R^2\to R is the canonical projection onto the first coordinate--why does this tell us D's connected. Conversely, if D is connected then \Gamma_f=h(D) where h: D\to R^2: x \mapsto (x,f(x))--why is h continuous and why does this tell us that \Gamma_f is connected?
    sorry, what is canonical projection ?
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