Cauchy Functional Equation

Cauchy's functional equation is the equation


It was proved by Cauchy in 1821 that the only continuous solutions of this functional equation from R into R are those of the form f(x)=kx for some real number k. 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 f(x) is nonnegative (or nonpositive) for sufficiently small positive x.

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.

The fifth of Hilbert's problems is a generalization of this equation.

See also

Abel's Functional Equation, Functional Equation, Hilbert's Problems

Portions of this entry contributed by José Carlos Santos

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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 f(x+y)=f(x)+f(y)." Math. Ann. 60, 459-462, 1905.

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Cauchy Functional Equation

Cite this as:

Santos, José Carlos and Weisstein, Eric W. "Cauchy Functional Equation." From MathWorld--A Wolfram Web Resource.

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