Question 36
Find the continuous and surjective functions $f : \mathbb{R} \to \mathbb{R}$ with the following two properties:
(i) $2f(1) = f(0) + f(2)$;
(ii) for any real numbers $a, b, c$, if $f(a), f(b), f(c)$ are terms of an arithmetic progression then $a, b, c$ are also terms of an arithmetic progression.
(i) $2f(1) = f(0) + f(2)$;
(ii) for any real numbers $a, b, c$, if $f(a), f(b), f(c)$ are terms of an arithmetic progression then $a, b, c$ are also terms of an arithmetic progression.