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.