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's 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 linear functions of the form f(x) = cx , where c is a constant. However, without such restrictions, there are many more complex and discontinuous solutions, highlighting the equation's significance in the study of functional analysis and real analysis.