Cauchy's functional equation is a well-known functional equation given by: \[ f(x + y) = f(x) + f(y) \] for all real numbers \(x\) and \(y\). This equation describes a function \(f\) that satisfies the property that the value of the function at the sum of two arguments is equal to the sum of the values of the function at each argument.
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Cauchy's functional equation asks for functions satisfyingThe identity must hold for every pair of real numbers, and a solution is an additive function. It says that combining two inputs before applying gives the same result as combining their outputs.[ref]
Remarks on the Cauchy functional equation and variations of it by Daniel Reem. Daniel Reem, Remarks on the Cauchy functional equation and variations of it, version 7, §1.1, Eq. (1) and the real domain
For example, satisfies the identity because . At and the two sides giveFor positive widths, a rectangle of height three makes this visible: cutting its width into and partitions its area into and . The same identity also holds for negative inputs, where is a signed output rather than an ordinary area.
The function f(x)=3x preserves addition: partitioning a rectangle and checking exact rational grid sums
. Details. Every function with a fixed real number solves Cauchy's functional equation. The surprising question is whether these are all the solutions. Rational homogeneity of an additive function forces for every rational number , but reaching an irrational number requires an additional argument. Continuity at just one point or being a bounded function on an interval of positive length supplies that argument; without such a condition, discontinuous additive functions exist.[ref]
Remarks on the Cauchy functional equation and variations of it by Daniel Reem. Reem, version 7, §1.1, printed p. 2: classical continuity and bounded-interval conditions and nonlinear solutions
This makes Cauchy's functional equation a useful test of the boundary between algebra and analysis: a rule about addition fixes all rational multiples, while a tiny amount of continuity fixes the entire real line. For example, the continuous exponential functional equation reduces to an additive function after taking a logarithm, so this result explains why its nonzero solutions are exponential functions.
Nice result on Lebesgue measurable required for uniqueness.