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Math Help - Example showing the Fixed Point Theorem is false with open intervals

  1. #1
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    Example showing the Fixed Point Theorem is false with open intervals

    Hi,
    I'm trying to find an example that the fixed point theorem will not be true if the closed interval [a,b] os replaced with the open interval (a,b).

    I'm using this definition:

    If f:[a,b]--->[a,b] is continuous then there exists a point c in [a,b] such that f(c)=c.

    Thanks!
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  2. #2
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    Opalg's Avatar
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    Take a=0 and b=1, so that we're dealing with the unit interval. Look for a continuous function on the (closed) unit interval whose only fixed points are at the endpoints of the interval. If you think in terms of the graph of the function, you want a function f with f(0)=0 and f(1)=1, whose graph lies entirely under (or over) the diagonal of the unit square (except at the endpoints). That shouldn't be too hard to find.
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  3. #3
    MHF Contributor Also sprach Zarathustra's Avatar
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    f(x)=e^x
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