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Math Help - Does my proof work?

  1. #1
    Junior Member
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    Does my proof work?

    The question is as follows. Suppose f is a continuous function on R, and
    f(f(x))=x for all x in R. Show that f(x) has at least one fixed point i.e.
    f(a)=a.

    Since f is continuous, by Extreme Value theorem, it attains a minimum m and a maximum M, for some values of x in R.

    m<=f(x)<=M
    f(m)<=f(f(x))=x<=f(M)

    By Intermediate Value Theorem, there exists a c in (f(m),f(M)) such that
    f(f(c))=c=f(c), so f has at least one fixed point on R.

    Does this work?

    Thanks in advance.
    Last edited by Hweengee; November 21st 2008 at 10:09 PM.
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  2. #2
    MHF Contributor kalagota's Avatar
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    Taguig City, Philippines
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    Quote Originally Posted by Hweengee View Post
    The question is as follows. Suppose f is a continuous function on R, and
    f(f(x))=x for all x in R. Show that f(x) has at least one fixed point i.e.
    f(a)=a.

    Since f is continuous, by Extreme Value theorem, it attains a minimum m and a maximum M, for some values of x in R. (you cannot talk about the extremums of f unless you are sure that f is bounded or the domain of f is bounded. you are only told that f is continuous..)

    m<=f(x)<=M
    f(m)<=f(f(x))=x<=f(M)

    By Intermediate Value Theorem, there exists a c in (f(m),f(M)) such that
    f(f(c))=c=f(c), so f has at least one fixed point on R.

    Does this work?

    Thanks in advance.
    ...
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  3. #3
    MHF Contributor
    Opalg's Avatar
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    Quote Originally Posted by Hweengee View Post
    The question is as follows. Suppose f is a continuous function on R, and
    f(f(x))=x for all x in R. Show that f(x) has at least one fixed point i.e.
    f(a)=a.
    Try doing this by contradiction. Suppose that f(x) is never equal to x. If it's sometimes greater than x and sometimes less than x then you can use the intermediate value theorem to get a contradiction. On the other hand, if it's always greater than x (or always less than x) then how could f(f(x)) ever be equal to x?
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