functional equations - A non-zero function satisfying $g(x+y) = g(x)g(y)$ must be positive everywhere

Let $g: \mathbf R \to \mathbf R$ be a function which is not identically zero and which satisfies the equation
$$
g(x+y)=g(x)g(y) \quad\text{for all } x,y \in \mathbf{R}.
$$

Show that $g(x)\gt0$ for all $x \in \mathbf{R}$.