Results 1 to 2 of 2

Math Help - is this correct?

  1. #1
    Member
    Joined
    Feb 2010
    Posts
    146

    is this correct?

    suppose that f: [a,b]->[a,b] is continuous. Prove that there is at least one fixed point in [a,b], that is, x such that f(x)=x


    This is what I have does it make sense?


    Since f is continuous, the function g(x) = f(x) - x is the difference of two continuous functions, hence g is continous on [a,b]

    If f(a) = a, then x=a is the fixed point. If not, then f(a) is in (a,b]. So f(a) > a, and g(a) > 0.

    If f(b) = b, then x=b is the fixed point. If not, then f(b) is in [a,b). So f(b) < b, and g(b) < 0.

    since g is continuous, and g(b) < 0 < g(a), the intermediate value theorem assures the existence of a point x = z in [a,b] at which g(z) = 0. but then f(z) - z = 0, so f(z) = z and z is a fixed point. (with x replaced with z)
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by tn11631 View Post
    suppose that f: [a,b]->[a,b] is continuous. Prove that there is at least one fixed point in [a,b], that is, x such that f(x)=x


    This is what I have does it make sense?


    Since f is continuous, the function g(x) = f(x) - x is the difference of two continuous functions, hence g is continous on [a,b]

    If f(a) = a, then x=a is the fixed point. If not, then f(a) is in (a,b]. So f(a) > a, and g(a) > 0.

    If f(b) = b, then x=b is the fixed point. If not, then f(b) is in [a,b). So f(b) < b, and g(b) < 0.

    since g is continuous, and g(b) < 0 < g(a), the intermediate value theorem assures the existence of a point x = z in [a,b] at which g(z) = 0. but then f(z) - z = 0, so f(z) = z and z is a fixed point. (with x replaced with z)
    Parfait!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 15
    Last Post: July 29th 2011, 02:39 AM
  2. Is this correct ?
    Posted in the Calculus Forum
    Replies: 3
    Last Post: July 16th 2010, 01:24 PM
  3. is this correct?
    Posted in the Math Topics Forum
    Replies: 1
    Last Post: April 24th 2009, 11:34 AM
  4. Is this correct
    Posted in the Pre-Calculus Forum
    Replies: 3
    Last Post: March 17th 2009, 06:43 AM
  5. is this correct?
    Posted in the Calculus Forum
    Replies: 2
    Last Post: July 2nd 2008, 12:00 AM

Search Tags


/mathhelpforum @mathhelpforum