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Homework Help: Continuous functions help

  1. Jan 27, 2010 #1
    I hope someone can help me wif this qnestion.

    Qn : Let f ; g be continuous functions from [0, 1] onto [0, 1]. Prove that there is
    x0 ∈ [0, 1] such that f (g(x0)) = g(f (x0)).

    Thanks in advance.
     
  2. jcsd
  3. Jan 27, 2010 #2
    Since both functions are onto [0,1], there are points a and b in [0,1] such that f(g(a)) = 0 and f(g(b)) = 1.

    Then we must have
    [tex]f(g(a)) - g(f(a)) \leq 0[/tex]
    and
    [tex]f(g(b)) - g(f(b)) \geq 0[/tex].

    The function h defined by [tex]h(x) = f(g(x)) - g(f(x))[/tex] is continuous. So...

    Maybe you can take it from here.
     
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