Let $f$ be defined on $\mathbb{R}$, and assume that there exists at least one point $x_{0} \in \mathbb{R}$ at which $f$ is continuous. Suppose also that, for every $x, y \in \mathbb{R}, f$ satisfies the equation

$$f(x+y)=f(x)+f(y)$$

Show that $f$ is continuous on $\mathbb{R}$.

Show that there exists a constant $c$ such that $f(x)=c x$ for all $x \in \mathbb{R}$.

Suppose that $g$ is a continuous function defined on $\mathbb{R}$ and that, for every $x, y \in \mathbb{R}$, $g$ satisfies the equation

$$g(x+y)=g(x) g(y) .$$

Show that if $g$ is not identically zero, then $g$ is everywhere positive. Find the general form of $g$.