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Math Help - show that f has a fixed point

  1. #1
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    show that f has a fixed point

    If f:[a,b]-->[a,b] is continuous, shwo that f has a fixed point; that is, show that there is some point x in [a,b] with f(x)=x.

    I'm really struglling with the continuous functions section with metric space, any help will be appreaciated!
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  2. #2
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    Re: show that f has a fixed point

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  3. #3
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    Re: show that f has a fixed point

    If f(a)= a we are done. If not, then, because f(a) must be in [a, b], f(a)> a.
    If f(b)= b we are done. If not, then, because f(b) must be in [a, b], f(b)< b

    Let \Phi(x)= f(x)- x.
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