# Does there exist a non-trivial function that satisfy the following relation?

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?

• 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. – fleablood Aug 9 at 0:26
• @fleablood $f(x,y)=0$ is the trivial solution. – Litun Aug 9 at 2:14
• What is the domain of $f$? – roxas3582 Aug 9 at 2:30
• @roxas3582 $\mathbb{R}^2$. – Litun Aug 9 at 2:33
• Yes, but $x-y$ is also a solution. But it isn't "trivial". – fleablood Aug 9 at 15:33

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?