# find all functions

#### bgbgbgbg

find all functions such that : for every x ,y in R
f(x)+f(y)=f(x+f(y))

#### Swlabr

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.

#### wonderboy1953

This is an interesting problem

I'm posting to track the progress on this problem.