Cauchy's Functional Equation is a mathematical equation that states if a function f satisfies the condition f(x + y) = f(x) + f(y) for all real numbers x and y , then f is a linear function. This means that the function can be expressed in the form f(x) = cx , where c is a constant. This equation is named after the French mathematician Augustin-Louis Cauchy, who studied it in the 19th century. The solutions to this equation are significant in various fields of mathematics, including analysis and functional equations, as they help in understanding the properties of functions and their behavior under addition.