By definition, a function is additive if $\displaystyle f(x+y) = f(x) + f(y)$ for all $\displaystyle x$, $\displaystyle y \in \mathbb{R}$.

How would you show that if $\displaystyle f$ is continuous at some point $\displaystyle x_0$, then it is continuous at every point of $\displaystyle \mathbb{R}$?