Cauchyyhtälö

Cauchyyhtälö, also known as Cauchy's functional equation, is a fundamental equation in mathematics. In its simplest form, it is expressed as f(x + y) = f(x) + f(y) for all real numbers x and y. This equation describes additive functions, meaning functions that preserve the operation of addition.

When considering the domain and codomain of the function to be the set of real numbers, solutions to the Cauchy functional equation can be surprisingly diverse. If we assume certain regularity conditions, such as continuity, monotonicity, or boundedness on an interval, then the only solutions are of the form f(x) = cx, where c is a constant. This linear form is intuitive, as it aligns with the concept of proportionality.

However, without these regularity assumptions, the existence of other, non-linear solutions becomes possible. These pathological solutions are often constructed using a Hamel basis for the real numbers as a vector space over the rational numbers. Such solutions are highly pathological and not encountered in typical applications.

The Cauchy functional equation appears in various areas of mathematics, including analysis, number theory, and abstract algebra. Its study provides insights into the nature of functions and the implications of different regularity conditions. Variations of the equation, such as Cauchy's exponential functional equation f(x+y) = f(x)f(y) or Cauchy's multiplicative functional equation f(xy) = f(x)f(y), are also important and have their own unique sets of solutions and properties. The additive form remains the most studied due to its direct connection to linearity and its foundational role in understanding function properties.