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

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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.

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Very nice. You've shown that if there IS a function that obeys this equation, then it must have the form that you've given. But to be a complete solution, you must ALSO show that all functions that HAVE this form obey the equation. – user22805 Aug 19 '12 at 4:07
@DavidWallace:Yes, you're right! So I must add a sentence reads"And it's easily checked that all those functions satisfy the equation." – Y.Z Aug 19 '12 at 4:17
or better still, add a sentence that actually checks it. – user22805 Aug 19 '12 at 4:19
@DavidWallace: I've checked that, and I leave this as an exercise for tangkhaihanh. :) – Y.Z Aug 19 '12 at 4:24

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