SCHWEITZER 1963 Problem 5

Let $H$ be a set of real numbers that does not consist of $0$ alone and is closed under addition. Further, let $f(x)$ be a real-valued function defined on $H$ and satisfying the following conditions: $$f(x)\leq f(y)\ \mathrm{if} x \leq y$$ and $$f(x+y)=f(x)+f(y) \;(x,y \in H)\ .$$ Prove that $f(x)=cx$ on $H$ , where $c$ is a nonnegative number. [M. Hosszu, R. Borges]