Is there a nonconstant continuous function $f:\mathbb{R}^2\rightarrow\mathbb{R}$ satisfying the functional equations $f(x,y)=f(x+y,y)=f(x,x+y)$?
If the answer is yes, can we characterize all solutions?
Edit: I think I got it. For every $(a,b)\in\mathbb{Z}^2$, the Eucledean GCD algorithms gives $f(a,b)=f(gcd(a,b),0)$. On the other hand, for every $(x,y)\in\mathbb{R}^2$, there are arbitrary close points $(\alpha a,\alpha b)$ with $a,b$ coprime integers and $\alpha\in\mathbb{R}$. So the answer is: such function must be constant.
I wonder if there are other approaches.