Determinate if exist a function $f:\Bbb R \rightarrow \Bbb R $ such that: $f(x+y)=max(xy,x)+min(xy,y)$ Determinate if it exists a function $f:\Bbb R \rightarrow \Bbb R $ such that: $$f(x+y)=\max(xy,x)+\min(xy,y)$$ My try was to define $(x,y)=(2,2)$ because in this way $x+y=xy$, and it didn't work, but I don't know if this is enough to determinate that this function does not exist. Any hints?

No, such function can not exist, because it would not be well-defined. For $r\in\mathbb{R}$ the function $f$ could possibly take infinite solutions, since you can write $r=x+y$ in an infinite amout of ways.

For example: $1=\frac12+\frac12$ and $1=2-1$.

Then $f(\frac12+\frac12)=\max(\frac14,\frac12)+\min(\frac14,\frac12)=\frac12+\frac14=\frac34$

Otherwise:

$f(2-1)=\max(-2,2)+\min(-2,-1)=2+(-2)=0$