Results 1 to 8 of 8

Math Help - Help finding this function?

  1. #1
    Member
    Joined
    Mar 2009
    Posts
    168

    Help finding this function?

    The idea of this "proof" is very easy to understand.

    Let f:[0,1]-->[0,1] be a continuous function. So you have a square one unit across and one unit tall. Draw a function inside of this box so that if you draw a horizontal line through the function anywhere, it will pass through the function 0 times or an even number of times (not infinitely many times).

    Remember, the function has to be continuous on [0,1].

    I thought for a while that this function doesn't exist, but my professor today confirmed it DOES exist. Has anyone ever heard of a theorem like this, and/or does anyone know a function that satisfies the criteria?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Aug 2007
    From
    USA
    Posts
    3,110
    Thanks
    2
    Why is that hard? Maybe I'm not understanding the requirements. I presume tangent would count as passing through zero times.

    Tell me what's wrong with f(x) = \frac{sin(2k\pi x)}{2}+\frac{1}{2} for k an integer and maybe we can understand the task.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Mar 2009
    Posts
    168
    Take k=1, for example.

    Then
    f(x) = \frac{sin(2\pi x)}{2}+\frac{1}{2}<br />

    So,
    f^{-1}(1)=.25

    In other words, the horizontal line y=1 only intersects the graph of f(x) one time, which is an odd number of times. So that function doesn't work. I need a function so that any horizontal line will cross it finitely many even times or zero times on [0,1], [0,1].

    I don't know what you meant about tangent being "zero times."

    Do you get what I mean now?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Mar 2009
    Posts
    168
    Also consider k=2. Then the horizontal line y=0.5 passes through the function 5 times...
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Mar 2009
    Posts
    168
    Anyone else have any ideas on this one?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor
    Joined
    Aug 2007
    From
    USA
    Posts
    3,110
    Thanks
    2
    You will have to make up your mind what you mean. The word "cross" is not well-defined. Do you mean "intersect"? If you do, then I agree with your negative assessment. If you do not mean "intersect", then you have not refuted my functions.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Member
    Joined
    Mar 2009
    Posts
    168
    Correct. They cannot intersect.

    I don't understand how this is possible, but my professor says it is.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Member
    Joined
    Mar 2009
    Posts
    168
    This is the solution, if anyone was interested.

    http://www.math.ufl.edu/~pilyugin/Pics/evenpreimage.pdf
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Finding a Cumulative Distribution Function and Density Function
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: November 17th 2011, 09:41 AM
  2. Replies: 14
    Last Post: October 7th 2011, 09:45 PM
  3. Finding probability function of moment generating function
    Posted in the Advanced Statistics Forum
    Replies: 6
    Last Post: July 4th 2011, 04:03 PM
  4. finding a function
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 30th 2011, 11:24 AM
  5. Finding the function
    Posted in the Calculus Forum
    Replies: 4
    Last Post: December 16th 2006, 04:15 AM

Search Tags


/mathhelpforum @mathhelpforum