The homogeneity and additivity properties together are called the superposition principle. It can be shown that additivity implies homogeneity in all cases where α is rational; this is done by proving the case where α is a natural number by mathematical induction and then extending the result to arbitrary rational numbers. If f is assumed to be continuous as well, then this can be extended to show homogeneity for any real number α, using the fact that rationals form a dense subset of the reals.
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