If $ f(1) = 0 $, then $ f(n) = 0 $ for all integers $ n $, and by additivity, $ f(r) = 0 $ for all rational $ r $. If $ f $ is continuous or linear, then $ f(x) = 0 $ or $ f(x) = x $. Without continuity, pathological solutions exist, but the multiplicative condition rules them out unless $ f(x) = 0 $ or $ f(x) = x $. - Groen Casting