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I need to construct a function $f(x,y)$ taking on continuous values between $0$ and $1$. $x$ and $y$ can take on any values (but their ranges may be restricted for convenience). This function should have the following properties:

  1. When $x$ large and $y$ large, $f$ is small
  2. When $x$ small and $y$ small, $f$ is small
  3. When $x$ large and $y$ small, $f$ is large
  4. When $x$ small and $y$ large, $f$ is large

I've been thinking of somehow using sines and cosines with different phase factors, but I don't know if this is the right approach. Any help is appreciated of course :)

Vladimir Khan
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1 Answers1

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It depends what do you mean by "x is small". If it means "close to zero", and thus x=100 and x=-100 both are not small, then this is works, for example:

$f(x,y) = (x^2-y^2)^2$

Example plot: http://www.wolframalpha.com/input/?i=%28x%5E2-y%5E2%29%5E2

Denis Zubo
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