Definition: A function $f : \mathbb{R} \to \mathbb{R}$ is said to be an Additive Function if for all $x, y \in \mathbb{R}$ we have that $f(x + y) = f(x) + f(y)$.

The following theorem gives us a nice characterization of continuity of additive functions. It tells us that the continuity of an additive function at a single point $x_0$ guarantees us the continuity of $f$ on all of $\mathbb{R}$.

 Theorem 1: Let $f : \mathbb{R} \to \mathbb{R}$ be an additive function. If $f$ is continuous at $x_0$ then $f$ is continuous on all of $\mathbb{R}$.
• Proof: Since $f$ is continuous at $x_0$ we have that:
(1)
\begin{align} \quad \lim_{x \to x_0} f(x) = f(x_0) \end{align}
• Let $x^* \in \mathbb{R}$. Since $f$ is additive we have that:
(2)
\begin{align} \quad f(x -x_0 + x^*) = f(x) - f(x_0) + f(x^*) \end{align}
• Therefore:
(3)
\begin{align} \quad \lim_{x \to x^*} f(x) &= \lim_{x \to x_0} f(x - x^* + x_0) \\ &= \lim_{x \to x_0} f(x) - f(x_0) + f(x^*) \\ &= f(x_0) - f(x_0) + f(x^*) \\ &= f(x^*) \end{align}
• So $f$ is continuous at every $x^* \in \mathbb{R}$, i.e., $f$ is continuous on all of $\mathbb{R}$.