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Calculus : Continuity

  1. Oct 19, 2014 #1
    1. The problem statement, all variables and given/known data
    If f(x) be a "continuous" function in interval [a,b] such that f(a)=b and f(b)=a, then prove that there exists at least one value "c" in interval (a,b) such that f(c)=c.

    Note: [a,b] denotes closed interval from a to b that is a and b inclusive. (a,b) denotes open interval from a to b that is excluding a and b.

    2. Relevant equations

    Concept of continuity.

    3. The attempt at a solution

    As function f(x) is continuous in [a,b] so graph of f(x) between x=a and x=b will be without any "break" and it covers value f(x) from a to b as well. Now as c lies between a and b i.e. a<c<b and f(b)=a and f(a)=b so there should be at least one solution of the equation f(x)=c. But how can we say that solution of equation f(x)=c is x=c ? How can I prove it ?

    Please help !

    Thanks in advanced... :)

    BTW, coming back after a long time!
     
  2. jcsd
  3. Oct 19, 2014 #2
    Consider the function ##g(x)=f(x)-x## on the interval ##[a,b]##.
     
  4. Oct 19, 2014 #3
    Why this function ? Question does not say it. Ummmm......

    (Thanks for quick reply)
     
  5. Oct 19, 2014 #4
    It's just a suggestion/hint. Note that ##f(c)=c## iff ##g(c)=0##.
     
  6. Oct 19, 2014 #5
    Please help me if i misunderstood. Question is asking us about f(x). What has g(x) to do with it and how will it answer the OP. And why we took g(x)=f(x)-x ? I do notice what you're saying though.
     
  7. Oct 19, 2014 #6
    If I were to say much more, it would become less of a hint and more of me telling you how to do the problem.

    Also, the suggestion was that you consider the function ##g##. I made you aware of it and implied that it was maybe pertinent to answering the problem. Now your job is to sit down and think about it for a bit. Maybe write down all of the facts that you can deduce about ##g## given what you know about ##f## and the relationship between ##a## and ##b##.
     
  8. Oct 19, 2014 #7
    I do know that between x=a to x=b the graph y=x intersect the curve f(x) at at least one point. At that coordinate is (x,x). But what i am not getting is that how is x=c necessarily at least at one point ?

    I have roughly sketched the figure.
     
  9. Oct 19, 2014 #8
    Are you allowed to call on the intermediate value theorem?
     
  10. Oct 19, 2014 #9

    Ray Vickson

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    Homework Helper

    You say "
    You say "I do know that between x=a to x=b the graph y=x intersect the curve f(x) at at least one point." How do you know that? That is exactly what you are trying to prove!
     
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