This is the **Cauchy Functional Equation**, whose solutions over $ \mathbbR $ are linear functions $ f(x) = kx $, assuming some regularity condition (like continuity, monotonicity, or boundedness). Since the problem does not restrict $ f $, but asks for the **number** of such functions, we must consider all additive functions from $ \mathbbR \to \mathbbR $. - go-checkin.com