The Cauchy Equation is a functional equation named after the French mathematician Augustin-Louis Cauchy. It typically takes the form f(x+y) = f(x) + f(y) , where f is a function and x and y are real numbers. This equation describes how the function behaves when adding two inputs together. Solutions to the Cauchy Equation can vary widely, but under certain conditions, such as continuity or boundedness, the solutions are linear functions of the form f(x) = cx , where c is a constant. This equation is fundamental in the study of functional analysis and has applications in various mathematical fields.