Cauchy functional equation

The Cauchy functional equation is a mathematical rule stating that for a function \(f\), applying it to the sum of two numbers is the same as adding the results of applying it to each number separately: \(f(x + y) = f(x) + f(y)\). This property characterizes functions called additive functions. Under typical assumptions like continuity, such functions are linear — essentially, they behave like straight-line equations (e.g., \(f(x) = kx\)). The equation is fundamental in understanding how functions can consistently distribute over addition, forming a basis for concepts like linearity in mathematics.