Let $f$ be a continuous homomorphism from $R \rightarrow R$.
Suppose $f(1)=r$ for some $r \in R$. Now we have to prove that for every $q \in Q$, $f(q)=rq$.
I figured out that for every integer $k$, $f(k)=kr$, even if it is less than zero (as if $k<0, f(1+1-1)=f(1), $and so $f(-1)=-r$). But if you split $q=k+c$, where $c<0$ and $c \in Q$ and $k\in Z$, then you can't figure out if $f(c)=cr$.
I really need help, I just can't figure it out!
Well, $r=f(1)=f(\frac{1}{k}+\frac{1}{k}+...+\frac{1}{k})=kf(\frac{1}{k}),$ implying $f(\frac{1}{k})=\frac{r}{k}$. Then, you should be able to follow your previous logic the rest of the way.