 A more special definition of linear function, not coinciding with the definition of linear map, is used in elementary mathematics (see below).Additivity alone implies homogeneity for rational α, since formula_1 implies formula_2 for any natural number "n" by mathematical induction, and then formula_3 implies formula_4.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.