Cauchy's power equation

Summary

Cauchy's power equation is the functional equation $f(xy) = f(x) f(y)$ for an unknown function $f \colon (0, \infty) \to \R$. It asks which functions turn products into products, that is, which functions are multiplicative. Its continuous solutions are the zero function and the power functions $f(x) = x^c$ with $c \in \R$. It is the fourth of Cauchy's basic equations, beside the additive, exponential and logarithmic ones.