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.