Additivity alone implies homogeneity for rational α, since f ( x + x ) = f ( x ) + f ( x ) implies f ( n x ) = n f ( x ) for any natural number n by mathematical induction, and then n f ( x ) = f ( n x ) = f ( m n m x ) = m f ( n m x ) implies f ( n m x ) = n m f ( x ) . The density of the rational numbers in the reals implies that any additive continuous function is homogeneous for any real number α, and is therefore linear.