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I am searching for a monotonically increasing and invertible function in $2$ variables. I know several monotonically increasing functions. This is also true for invertible functions. But I am searching for a function which is both monotonically increasing and invertible in $2$ variables.

Eg. a function of the form: $z=f(x,y)$ , here x<=y such that we can get $x=f^{-1}(z)$ and $y=f^{-1}(z)$ Also, if $x_1 \lt x_2$ then $z_1 \lt z_2$. In case $x_1=x_2$ then if $y_1\lt y_2$ then $z_1\lt z_2$.

Please help me figure out some function of this type.

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    You could get $(x,y)=f^{-1}(z)$, but not separately. Also you don't give any indication about the function domain, maybe you intend $\mathbb{R}^2$?2012-07-29
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    @enzotib First and foremost thanks for replying. But I want separately.2012-07-29
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    I would guess that any invertible function $\mathbb{R}^2 \to \mathbb{R}$ must be a bit odd?2012-07-29
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    @copper.hat So sorry, I did not get what u said.2012-07-29
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    The short answer is: such functions don't exist.2012-07-29
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    @JannatArora: Informally, you are trying to map two dimensions to one in a way that you can invert the function. The only time I have seen this sort of thing in in cardinality arguments. In addition you are looking for monotonicity in each variable separately. So, if such a function existed, it would be a strange beast.2012-07-29
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    copper.hat ya...but i m in dire need of such a function2012-07-29

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