67th Romanian Mathematical Olympiad

Inductively we obtain $f(x+r)\le f(x)+r$, for any $x\in \mathbb{R}$ and $r\in \mathbb{Q}$.

The continuity of $f$ and the density of $\mathbb{Q}$ in $\mathbb{R}$ give $f(x+y)\le f(x)+y$, for all $x\in \mathbb{R}$ and $y\in \mathbb{R}$. We get thus the functions defined by $f_a(x)=x+a$, for $x\in \mathbb{R}$ where the parameter $a$ runs over $\mathbb{R}$. It is obvious that all these functions verify the hypothesis.