It was proved by Cauchy in 1821 that the only continuous solutions of this functional equation from into are those of the form for some real number . In 1875, Darboux showed that the continuity hypothesis could be replaced by continuity at a single point and, five years later, proved that it would be enough to assume that is nonnegative (or nonpositive) for sufficiently small positive .
In 1905, G. Hamel proved that there are non-continuous solutions of the Cauchy functional equation using Hamel bases. Every non-continuous solution is necessarily non-measurable with respect to the Lebesgue measure.
Aczél, J. Lectures on Functional Equations and Their Applications. New York: Academic Press, 1966.Broggi, U. "Sur un théorème de M. Hamel." Enseignement Math.9, 385-387, 1907.Cauchy, A. L. Cours d'Analyse de l'Ecole Royale Polytechnique. Chez Debure frères, 1821.Darboux, G. "Sur la composition des forces en statique." Bull. Sci. Math.9, 281-299, 1875.Darboux, G. "Sur le théorème fondamental de la Géométrie projective." Math. Ann.17, 55-61, 1880.Hamel, G. "Eine Basis aller Zahlen und die unstetigen Lösungen der Funktionalgleichung ." Math. Ann.60, 459-462, 1905.
into 
are those of the form 
for some real number 
. In 1875, Darboux showed that the continuity hypothesis could be replaced by continuity at a single point and, five years later, proved that it would be enough to assume that 
is nonnegative (or nonpositive) for sufficiently small positive 
. 
." Math. Ann. 60, 459-462, 1905.