= Continuous exponential functional equation
{title2=$g(x+y)=g(x)g(y)$}
A continuous real <function> satisfying $g(x+y)=g(x)g(y)$ is either identically zero or $g(x)=e^{cx}$ for a real constant $c$. If it is nonzero somewhere, $g(0)=1$ and $g(x)g(-x)=1$, so it never vanishes. The identity $g(x)=g(x/2)^2$ then makes it everywhere positive. Its real logarithm is a continuous <additive function>, hence linear by <additive function continuous at one point is linear>. Exponentiating proves the classification, and each listed <function> satisfies the equation.
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