1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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]
    [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.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook