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Math Help - Borsuk-Ulam Theorem 1 dimensional

  1. #1
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    Borsuk-Ulam Theorem 1 dimensional

    Let f:S^1\to \mathbb{R} be a continuous map. Show there exists a point x of S^1 such that f(x)=f(-x).

    It says to define g:S^1\to \mathbb{R} by g(x)=f(x)-f(-x).

    g(x)=f(x)-f(-x)=-(f(-x)-f(x))=-g(x)

    By the Intermediate Value Theorem, there exist x_0\in S^1 such that g(x_0)=0\Rightarrow f(x)=f(-x).

    Here is my question: Why did we need to define a new function g?
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  2. #2
    Senior Member roninpro's Avatar
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    Re: Borsuk-Ulam Theorem 1 dimensional

    You defined the new function g so that you could apply the Intermediate Value Theorem in a straightforward manner.

    Did you have something else in mind?
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