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I try to define a two-variable function $f(x,y) \ge 0$ with both $x,y \in [0,1]$ which satisfies the following constraints:

  1. $f$ is increasing in both $x$ and $y$
  2. if $x=0$ then $f(x,y)=0$
  3. if $y=0$ then $f(x,y)=0$ only if $x=0$

Any suggestion?

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up vote 5 down vote accepted

Would the function $f(x,y)=x(y+1)$ work?

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thanks for the suggestion. Is it possible to not to involve some arbitrary constant $\alpha \gt 0$ (such as $\alpha = 1$ here)? The inconvenience is that I have to justify for the particular choice of $\alpha$ here. – skyork Feb 19 '12 at 18:05
@skyork: $f(x,y)=x\cdot g(y)$ should work for any function $g$ that is increasing on $[0,1]$ with $g(0)>0$. I'm not sure that any particular choice of $g$ is any less arbitrary that $g(y)=y+1$, though. – Isaac Feb 19 '12 at 18:08

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