The Additive Cauchy equation is a functional equation that can be expressed as f(x + y) = f(x) + f(y) for all real numbers x and y . This equation describes a property of functions that are additive, meaning the function's output for the sum of two inputs is equal to the sum of the outputs for each input separately. Solutions to the Additive Cauchy equation are typically linear functions of the form f(x) = cx , where c is a constant. This equation is significant in the field of mathematics, particularly in the study of functional equations and real analysis, and it has connections to topics like linear algebra and measure theory.