Cauchy's exponential equation

Remarks

Example: $f(x) = e^{3x}$ is a solution, since $e^{3(x + y)} = e^{3x} e^{3y}$. Non-example: $f(x) = 1 + x$ is not a solution, since $f(1 + 1) = 3$ while $f(1) f(1) = 4$. Every solution satisfies $f(x) = f(x/2)^2 \ge 0$. If $f(x_0) = 0$ for one $x_0$, then $f(x) = f(x - x_0) f(x_0) = 0$ for all $x$. Otherwise $f > 0$ and $\ln f$ is an additive function, so the solutions are $f = 0$ and $f = e^{A}$ with $A$ additive. The continuous solutions are $f = 0$ and $f(x) = a^x$ with $a > 0$.