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Is it possible to find a monotonic bijection that takes as input 2 functions with domain non-negative real numbers and outputs a real value?

This is because I was wondering whether it was possible to combine 2 performance measures and maximize their combination.

Cap_F
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  • "monotonic bijection that takes..." I suspect no and suspect this is already asked here before. "If it was possible to combine 2 performance measures" If this is in a programming context, not in a bijective way without information loss. Recall that there are only finitely many possible numbers representable by a float or whichever standard data type you may be using, implying by pigeonhole principle that two floats being combined into one float would be mapping a space of $n^2$ possibilities into a space of only $n$ possibilities can not be bijective. – JMoravitz Jun 02 '23 at 12:40
  • See Prove or disprove there exists a continuous bijection from $\Bbb R^2$ to $\Bbb R$. It should be equivalent to your question. – JMoravitz Jun 02 '23 at 12:43
  • Thanks, that makes sense. I was just considering whether it was possible to both minimize the mse and maximize the correct sign of the prediction (a sort of accuracy measure) in a regression setting. I guess it is not feasible precisely because of your reasoning. – Cap_F Jun 02 '23 at 12:57
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    What does it mean for a function of two arguments to be monotone? – Joshua Tilley Jun 02 '23 at 13:10
  • @JMoravitz, the OP did not mention continuity, though I suppose they might only be interested in that case. – Joshua Tilley Jun 02 '23 at 13:11
  • @JoshuaTilley if it were monotonic and bijective, wouldn't that imply continuity? – JMoravitz Jun 02 '23 at 13:27
  • What does it mean for it to be monotone, I expect that if $X\geq x$ and $Y\geq y$ then $f(X,Y)\geq f(x,y)$ – JMoravitz Jun 02 '23 at 13:28
  • It has two arguments - for a function with two arguments what is even meant by monotonic? – Joshua Tilley Jun 02 '23 at 13:28

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