Question 40
Find all functions $f : \mathbb{R} \to \mathbb{R}$ that satisfy the following conditions:
(a) $x + f(y + f(x)) = y + f(x + f(y))$, $\forall x, y \in \mathbb{R}$
(b) The set $I = \left\{\frac{f(x)-f(y)}{x-y} \mid x, y \in \mathbb{R}, x \neq y\right\}$ is an interval. (means for any two numbers $u, v \in I$ with $u \le v$, we have $\forall w \in [u, v] : w \in I$)
(a) $x + f(y + f(x)) = y + f(x + f(y))$, $\forall x, y \in \mathbb{R}$
(b) The set $I = \left\{\frac{f(x)-f(y)}{x-y} \mid x, y \in \mathbb{R}, x \neq y\right\}$ is an interval. (means for any two numbers $u, v \in I$ with $u \le v$, we have $\forall w \in [u, v] : w \in I$)