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Homework Help: Question about a continuous function on R2

  1. Sep 23, 2010 #1
    1. The problem statement, all variables and given/known data

    Let S denote a unit circle centered at origin in xy plane, and
    f is a continuous function that sends S to R (no need to be 1 to 1 or onto).
    show that there's (x, y) such that f(x, y) = f(-x, -y)

    2. Relevant equations

    have a feeling it has something to do with theorems related to topology.


    3. The attempt at a solution

    by re-writing S as (cos z, sin z) for 0 <= z < 2 pi
    i can reduce f to a function of one variable, f(z), such that
    f(z) = - f(z + pi), but then I'm stuck what to do after that.

    (or maybe I shouldn't have reduced f to a function of one variable in the first place?)
     
  2. jcsd
  3. Sep 23, 2010 #2

    Dick

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    Reducing it to one variable is fine. And, yes, (x(z+pi),y(z+pi)) is then (-x(z),-y(z)) but how does that tell you anything about the value of f? Try considering g(z)=f(z)-f(z+pi). Can g(z) be nonzero for all values of z? This is more about continuous functions than it is about topology.
     
  4. Sep 23, 2010 #3
    yeah, i did that, but i failed to show that g(z) must be zero for some value of z, and
    that was exactly the problem I had.
    If I can show that if g(z) > 0 for some value of z, then g(x) < 0, for some value of x that's
    not equal to z, then I can apply an intermediate value theorem to solve it, but
    I also forgot how to apply an intermediate value theorem in 2 dimensional case.

     
  5. Sep 23, 2010 #4

    Dick

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    You aren't in the two dimensional case anymore, you reduced it to a function of one variable, z, didn't you? Start thinking in one dimension.
     
  6. Sep 23, 2010 #5
    true. but with all due respect, you're not adding anything to what i've done so far.
    my problem was that I am not quite sure how to prove the existence of z, x such that
    they are not equal to each other, and
    g(z) > 0 and g(x) < 0 (or g(z) <= 0 and g(x) > 0)
    so that I can apply intermediate value theorem.
     
  7. Sep 23, 2010 #6

    Dick

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    Alright, so you need a stronger hint. How is g(z) related to g(z+pi)?
     
  8. Sep 23, 2010 #7
    you know what? i solved it. you don't have to continue to reply to this.
    plus, it's not a "hint" if it's something that I already proved on my own.
     
  9. Sep 23, 2010 #8

    Dick

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    Very welcome.
     
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