CauchyGleichungen

CauchyGleichungen, also known as Cauchy's functional equations, are a class of functional equations named after the mathematician Augustin-Louis Cauchy. They are equations that relate a function of a sum to the sum of the functions. The most fundamental Cauchy equation is f(x + y) = f(x) + f(y) for all real numbers x and y.

If the function f is assumed to be continuous, then the only solutions to f(x + y) = f(x) + f(y) are of the form f(x) = cx, where c is a constant. However, if no continuity assumption is made, there exist other, more complex solutions that are not linear. These non-linear solutions are constructed using a Hamel basis for the real numbers as a vector space over the rational numbers, and they exhibit highly pathological behavior, such as being nowhere continuous and unbounded on any interval.

Beyond the additive Cauchy equation, other related functional equations are also sometimes referred to as Cauchy's functional equations. These include:

f(x + y) = f(x)f(y), whose continuous solutions are of the form f(x) = a^x.

f(xy) = f(x) + f(y), whose continuous solutions for positive x, y are of the form f(x) = c ln(x).

f(xy) = f(x)f(y), whose continuous solutions for positive x, y are of the form f(x) = x^c.

These equations and their solutions are foundational in the study of functional analysis and have applications in various areas of mathematics and physics, particularly in contexts where linearity or specific growth properties are investigated. The existence of non-linear solutions for the additive equation highlights the importance of regularity conditions like continuity when solving functional equations.