Pexider equation

Summary

The Pexider equation is the functional equation $f(x + y) = g(x) + h(y)$ with three unknown functions $f, g, h \colon \R \to \R$. It generalizes Cauchy's additive equation, which is the case $f = g = h$. Its solutions are $f = A + a + b$, $g = A + a$ and $h = A + b$, where $A$ is an additive function and $a, b$ are constants. Equations of this kind, with several unknown functions in the place of one, are called Pexiderized forms of the classical equations.