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Let $f: \mathbb{R} \to \mathbb{R}$ $$f(x^2+y)=f(x)+f(y^2)$$

How do I find all functions that fulfill this equation?

I tried to just write many equalities but it just doesnt help.

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And what is the question about that equation? – Gerry Myerson Jun 3 '11 at 6:03
You might get a more complete response at $$ $$ $$ $$ where they specialize in contest type problems $$ $$ – Will Jagy Jun 3 '11 at 6:06
Hi Vadiklk, By enclosing LaTeX in $-signs (e.g. writing $x^2$) you get the formulas typeset (e.g. $x^2$). I edited both your questions in that respect and made the title a little bit more descriptive. I hope that's fine with you. If not, you can click on edited xx secs/mins/hours ago above my name and roll back to your original version. – t.b. Jun 3 '11 at 6:12
What are the domain and codomain of your function? – N. S. Jun 3 '11 at 6:14
up vote 5 down vote accepted

You can solve this pretty immediately by looking at the cases $x=0$, $y=0$, and $x^2+y=0$. (The only possibility is the constant zero function.)

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thanks alot, you helped me really much! – Vadiklk Jun 3 '11 at 7:02

I would try to see how it behaves at "special points" like +-1, 0 and some others, and see how it can behave.

For example, if you let $y=0$ this becomes $f(x^2) = f(x)$.

Then for $x=y=1$ you get $f(2) = 2f(1)$.

For $x=1, y=-1$ you get $f(0) = 2f(1)$ also...

Try a few more combinations and you'll get enough constraints to define $f$, or reach a contradiction.

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