Start by noticing that f(0)=0. Indeed, putting a=b=0, one gets f(0)=2f(0), hence f(0)=0.

Now put a=x, b=h:

f(x+h)-f(x)=f(h)=f(h)-f(0)

Divide by h to get

(f(x+h)-f(x))/h=(f(h)-f(0))/h

Take the limit h->0 to finally get

f'(x)=f'(0).

This means that f'(x)=a is a constant: f(x)=cx+d, but d must be zero in order to the first relation to hold.

Hence f(x)=cx. Those are called linear functions.