It's known that the solutions to the Cauchy's Functional Equation:
$$f(x+y) = f(x) + f(y)$$
are of the form $f(x) = cx$, where $c$ is a constant. I've read that this situation takes place when the function $f(x)$ is continuous. But I would like to know if all the solutions to the Cauchy's Functional Equation can be described by using the theorem on the existence of Hamel bases.