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I am looking for a function $f$ that satisfy the following relation $f(y+\alpha*f(x,y),y)=\alpha*f(x,y)$, where $\alpha\in [0,1]$. The trivial one that satisfy this relation is $f(x,y)=x-y$. I tried with several other examples, but failed to find another example. Does there exist any non-trivial function that satisfy the above relation?

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  • $\begingroup$ Nitpick. That function may be simple, or easy, or obvious, but it is not "trivial". "Trivial" has a very specific meaning in mathematics and the function is $f(x,y)=0$ which... now that you mention it, does work. $\endgroup$ – fleablood Aug 9 at 0:26
  • $\begingroup$ @fleablood $f(x,y)=0$ is the trivial solution. $\endgroup$ – Litun Aug 9 at 2:14
  • $\begingroup$ What is the domain of $f$? $\endgroup$ – roxas3582 Aug 9 at 2:30
  • $\begingroup$ @roxas3582 $\mathbb{R}^2$. $\endgroup$ – Litun Aug 9 at 2:33
  • $\begingroup$ Yes, but $x-y$ is also a solution. But it isn't "trivial". $\endgroup$ – fleablood Aug 9 at 15:33
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For general ways to attack such kind of functional equations, take a look at Evan Chen's Introduction to Functional Equations and Monsters. Introductory, oriented at Math Olympiad competitions, it is quite understandable and thorough (as far as it goes).

First question: Does $f$ have to satisfy your equation for all $\alpha$ (what range of values?) or are you searching for functions that satify the equation for some specific values of $\alpha$?

Next, fool around a bit. What can you deduce for specific $x, y$? Zero, $x = - y$, ...? You have found a solution. Does a multiple of it work? It is a linear combination of $x, y$. Any others?

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