4

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.

Luke
  • 41
  • 1
    I'm not getting the result from the last sentence. $(x,y) \in \mathbb{R}^2$ but then $x, y$ are coprime integers? – MT_ Apr 19 '16 at 19:39
  • I've not got any better method than that. I think it's quite neat, actually. – Patrick Stevens Apr 19 '16 at 21:48
  • 1
    @Soke The idea is that there is a sequence of points of the form $(\alpha a, \alpha b)$ that converges to $(x,y)$ with $\alpha \to 0$: by continuity $f(\alpha a, \alpha b) \to f(x,y)$. The value of $f$ at each point is equal to $f(\alpha,0)$ so they also converge to $f(0,0)$ by continuity. – Erick Wong Apr 20 '16 at 02:30

0 Answers0