Results 1 to 2 of 2

Math Help - Proof using the Intermediate Value Theorem

  1. #1
    gmencl
    Guest

    Proof using the Intermediate Value Theorem

    A fixed point of a function f is a number c in its domain such that f(c) = c. (the function doesn't move the input value c; it stays fixed.) Suppose that f is a continuous function with domain [0,1] and range contained in [0,1]. Prove that f must have at least one fixed point.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by gmencl View Post
    A fixed point of a function f is a number c in its domain such that f(c) = c. (the function doesn't move the input value c; it stays fixed.) Suppose that f is a continuous function with domain [0,1] and range contained in [0,1]. Prove that f must have at least one fixed point.
    Wow, what a beautify theorem. I never really appreciated fixed points theorems until now.

    Define a function on the closed interval [0,1]
    g(x)=f(x)-x
    We note that since f(x) and x are continous on [0,1] so too g(x) is continous on [0,1].
    We note that g(0)=f(0)-0=f(0)
    And g(1)=f(1)-1
    Here is the crucial steps:
    f(0)=>0 by the defintion of f(x).
    If f(0)=0 then the proof is complete, that is c=0.
    Thus, there is not harm in assuming f(0)>0
    f(1)<=1 thus, f(1)-1<=0
    If f(1)-1=0 the proof is complete, that is c=1.
    Thus, there is no harm in assuming f(1)-1<0
    Since g(x) is continous by intermediate value theorem, [zero is between the f(0) and f(1)-1]
    There is a point c in [0,1] such that, g(c)=0
    But, g(c)=f(c)-c thus, f(c)=c.
    Q.E.D.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Intermediate value theorem proof
    Posted in the Calculus Forum
    Replies: 1
    Last Post: August 13th 2011, 11:17 PM
  2. Possibly an intermediate value theorem proof, but I'm not sure.
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: July 4th 2011, 10:08 AM
  3. Proof of intermediate value theorem for a particular case
    Posted in the Differential Geometry Forum
    Replies: 11
    Last Post: April 26th 2011, 07:37 AM
  4. intermediate value theorem/rolle's theorem
    Posted in the Calculus Forum
    Replies: 6
    Last Post: December 8th 2007, 02:55 PM
  5. Proof with Intermediate Value Theorem
    Posted in the Calculus Forum
    Replies: 2
    Last Post: March 28th 2007, 08:53 PM

Search Tags


/mathhelpforum @mathhelpforum