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Math Help - Pathwise connected proof

  1. #1
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    Pathwise connected proof

    Ok, so I would like to know how to prove the following proof. If anyone has any insight I would greatly appreciate it.

    Suppose D is pathwise connected and \rightarrow R" alt="f\rightarrow R" /> is continuous. Prove that f(D) is an interval.

    Is it enough to create a function that is continuous and then show that the function lives on an interval?

    For example, \phi (t)=(1-t)\alpha +t\beta where \phi (0)=\alphaand \phi (1)=\beta . Then show that the function lives on the interval [0,1]?
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  2. #2
    Senior Member roninpro's Avatar
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    You have to show that this works for any arbitrary function. Picking a particular function (as you did) will not suffice.

    You are really being asked to show two things:

    1) If D is pathwise connected, X is a space, and \to X" alt="f\to X" /> is continuous, then f(D) is pathwise connected.

    2) A subset of \mathbb{R} is pathwise connected if and only if it is an interval.

    Give it a try.
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  3. #3
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    I have the proof of part 2 in my notes that I obtained from previous guidance. So I guess part one is what I struggle with. Its not hard for me to believe that it is true but I don't know how to get started. Maybe a proof by contradiction?
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  4. #4
    Senior Member roninpro's Avatar
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    Recall the definition of a pathwise connected set X: we say that X is pathwise connected if for points a,b\in X, there exists a continuous function \gamma:[0,1]\to X such that \gamma(0)=a and \gamma(1)=b. Our goal will be to produce such a function for our X.

    You have two important facts: D is pathwise connected, so given a,b\in D, there exists a continuous function \gamma_D:[0,1]\to D such that \gamma(0)=a and \gamma(1)=b; and \to X" alt="f\to X" /> is continuous. How can you come up with \gamma:[0,1]\to f(X) connecting two points?
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  5. #5
    MHF Contributor Drexel28's Avatar
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    Not to butt in, but I feel as though this is simpler. We (and by we I mean you, zebra, and I) have already proven that a path connected space is connected. So, D is connected and so by continuity f(D) is connected. And, surely you know every connected subspace of \mathbb{R} is an interval.
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