The Cauchy Functional Equation is a mathematical equation that expresses a relationship between a function and its inputs. It is typically written as f(x + y) = f(x) + f(y) for all real numbers x and y . This equation suggests that the function f is additive, meaning that the value of the function at the sum of two inputs is equal to the sum of the function values at those inputs. Solutions to the Cauchy Functional Equation can vary widely depending on additional conditions imposed on the function. If f is assumed to be continuous, then the only solutions are of the form f(x) = cx , where c is a constant. However, without such conditions, there can be many more exotic solutions, including those that are not well-behaved or discontinuous.