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Continuity Proof

  1. Mar 14, 2013 #1
    1. The problem statement, all variables and given/known data

    The problem statement and proof can be found here. The proof continues after this, but I only have a question about the beginning of the proof.

    2. Relevant equations

    NA

    3. The attempt at a solution

    My question is simply this:

    Every proof I find for this problem starts assuming the inequality [itex]0≤(x-y^2)^2[/itex] and I cannot figure out why. I'm sure this is a simple matter, but any explanation would be appreciated.

    Thanks!
     
  2. jcsd
  3. Mar 14, 2013 #2

    jbunniii

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    The square of a real number can't be negative, right?
     
  4. Mar 14, 2013 #3
    Yes, indeed, I see that it works. But was this first step just conjured out of experience and a keen eye, or did something in the problem suggest it specifically?

    Thanks!
     
  5. Mar 14, 2013 #4

    jbunniii

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    Probably experience and a keen eye. If you play around with inequalities often enough, you start to recognize things like the fact that the numerator of
    $$\frac{xy^2}{x^2 + y^4}$$
    can be used to "complete the square" in the denominator, i.e.
    $$x^2 + y^4 + 2xy^2 = (x + y^2)^2$$
    and
    $$x^2 + y^4 - 2xy^2 = (x - y^2)^2$$
    If you don't notice this trick, there are usually other ways to find a bound.
     
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