Cauchy's exponential equation

Summary

Cauchy's exponential equation is the functional equation $f(x + y) = f(x) f(y)$ for an unknown function $f \colon \R \to \R$. It asks which functions turn sums into products. Its continuous solutions are the zero function and the exponential functions $x \mapsto a^x$ with $a > 0$. It is one of the four basic equations of Cauchy, together with the additive, logarithmic and power equations.