Observe that since f(x+x)=f(x)+f(x)=2f(x)
for any x, and any integer you have
now pick a rational number p/q q non zero. we have that
then multiply by q/q and use the first result
So we have proved that for every rational number a.
In the next the continuity is crucial, let x be any real number, and sequence of rationals converging to x .
so since f is continuous,
hence where K=f(1)
Edit: The above works only where f is continous so by Lusin's theorem f(x)=Kx on F where m(F) is small.
Im not sure how to prove that f is linear everywhere... let me think