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Math Help - Check 1 more Real

  1. #1
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    Check 1 more Real

    Suppose we have a continuous function f that is on the closed interval [0,1] and has a range that's contained in [0,1] as well. Prove f has to have a fixed point (ie: show f(x) = x for at least 1 value of x \in [0,1].

    SOLUTION:

    We are done if f(0) = 0 or f(1) = 1

    What about if f(0) > 0 and f(1) < 1.

    Let g(x) = x - f(x), then we know that g is continuous and further g(0) < 0, g(1) > 0. By the Intermediate Value Theorem (IVP), there exists a c \in (0,1) with g(c) = 0. That is, f(c) = c. And hence the proof is complete.
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  2. #2
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    Yes, that proof works nicely.
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