real analysis - $f(a+b)=f(a)+f(b)$. Prove that $f(x)=Cx$, where $C=f(1)$

A question from Introduction to Analysis by Arthur Mattuck:

Suppose $f(x)$ is continuous for all $x$ and $f(a+b)=f(a)+f(b)$ for all $a$ and $b$. Prove that $f(x)=Cx$, where $C=f(1)$, as follows:

(a)prove, in order, that it is true when $x=n, {1\over n}$ and $m\over n$, where $m, n$ are integers, $n\ne 0$;

(b)use the continuity of $f$ to show it is true for all $x$.

I can show the statement is true when $x=n$. As for $x={1\over n},{m\over n}$, I don't know how.