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Fixed Point in Continuous Set

  1. Nov 20, 2008 #1
    1. The problem statement, all variables and given/known data
    Suppose f:[a,b] [tex]\rightarrow[/tex] [a,b] is continuous. Prove that there is at least one fixed point in [a,b] - that is, x such that f(x) = x.


    2. Relevant equations



    3. The attempt at a solution
    I was going to try something with the IVT, but then I realized I wasn't sure what they meant by a fixed point much less how to solve this problem. Any help would be appretiated.
     
  2. jcsd
  3. Nov 20, 2008 #2
    Hi Angelpsymon,

    It says what a fixed point is in the problem statement: "x such that f(x) = x." You are absolutely correct in thinking to apply the intermediate value theorem. Hint: since f maps into [a,b], we must have that [tex]f(a)\geq a[/tex] and [tex]f(b)\leq b[/tex].
     
  4. Nov 20, 2008 #3

    Dick

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    As Unco suggested, maybe without spelling it out completely, apply the IVT to f(x)-x.
     
  5. Nov 20, 2008 #4
    Apologies, Dick, I certainly didn't mean to do so.
     
  6. Nov 20, 2008 #5

    Dick

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    You don't HAVE to spell it out completely. Hints are enough. I apologize if I spoiled your hint and made it too obvious. I was just saying how to apply the IVT.
     
  7. Nov 22, 2008 #6
    Alright, I think that I got it now. Thanks a lot guys.
     
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