# Find all continuous functions $f:\mathbb{R}\to\mathbb{R}$, that satisfy $f(xy)=f(\frac{x^2+y^2}{2})+(x-y)^2$

Find all continuous functions $f:\mathbb{R}\to\mathbb{R}$, that satisfy $f(xy)=f(\frac{x^2+y^2}{2})+(x-y)^2$

-
"Welcome to math.SE: since you are new, I wanted to let you know a few things about the site. Titles should be informative. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. If this is homework, please add the homework tag; people will still help, so don't worry. Also, many find the use of imperative ( "Find") to be rude when asking for help; please consider rewriting your post." –  Pedro Tamaroff Aug 19 '12 at 3:58

Let $x=0$ first, we see $f(0)=f(\frac{y^2}{2})+y^2$, hence $f(x)=f(0)-2x$ if $x\ge 0$. Then let $y=-x$, we get $f(-x^2)=f(x^2)+4x^2$, hence $f(x)=f(-(-x))=f(-x)+4(-x)=f(0)-2x$ when $x\le 0$.
And it's easily checked that all those functions satisfy the equation. So all you want is $f(x)=-2x+c$ for any constant c.