Here's the question:
Let $f\colon \mathbb{R} \to \mathbb{R}$ be continuous which is also an additive homomorphism, that $f(x+y)=f(x)+f(y)$ for all $x.y \in\mathbb{R}$. Then show that $f(x)=\lambda x $ where $ \lambda = f(1)$.
Truth to be told, I have no idea how to solve this problem. Here's what the authors have discussed before this Exercise.
Let $J\subset \mathbb{R}$. Let $f\colon J\to \mathbb{R}$ be a function and $a\in J$. We say that $f$ is continuous at $a$ if for every sequence $(x_n)$ in $J$ with $\lim x_n =a$, we have $\lim f(x_n) = f(a)$. We also say that $f$ is continuous on $J$ if it is continuous at every point $a\in J$.
They discuss few examples before dropping this exercise. $\epsilon- \delta$ definition of continuity has not been defined yet and this is the first section of the chapter.
I'd appreciate it if you give me some hints instead of solving it entirely.
Hint: Use this answer and the density of $\Bbb Q$ in $\Bbb R$.