find all functions such that : for every x ,y in R
f(x)+f(y)=f(x+f(y))
Interesting question. Is there any context? For example are these linear functions? Or all functions?
$\displaystyle f_n(x) = x+n$ would be one, for all $\displaystyle n \in \mathbb{R}$ (this includes the identity function), as would the zero function, $\displaystyle f(x)=0$.
Now, if the question was about linear bijections then there is only one such function, which would be a nice question. However, this question is much too vague. I mean, the function,
$\displaystyle g_n(x) = \left\{
\begin{array}{lr}
x & : x \in \mathbb{Q}\\
x+n & : x \notin \mathbb{Q}
\end{array}
\right.$
is a function of this form. And it's hideous.