CauchyGleichung

CauchyGleichung refers to the Cauchy functional equation: a function f from the real numbers to the real numbers that satisfies f(x+y) = f(x) + f(y) for all real x and y. Named after Augustin-Louis Cauchy, the equation expresses additivity and is a basic example of a functional equation in analysis. It can be viewed as a homomorphism between the additive groups (R, +) and itself, and, more generally, on any abelian group.

The general set of solutions consists of all additive functions. If a function f is additive on R, it is determined by its values on a Hamel basis of R as a vector space over the rationals, and every such choice yields a solution. However, without additional regularity conditions, there are many pathological additive functions that are not linear in the usual sense.

With extra regularity, the situation changes: if f is continuous at a point, bounded on any interval, or measurable, then f(x) = c x for some constant c, where c = f(1). Thus every regular additive function is linear.

In practice, Cauchy’s equation serves to study linearity properties without assuming smoothness. Extensions include considering the equation on other domains (such as vector spaces over Q or other fields) and examining the set of all homomorphisms from one additive group to another.

Historically, the equation has played a foundational role in the development of functional equations and the understanding of additive structures in real analysis.