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.