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