Let a function $f \colon \mathbb{R} \to \mathbb{R}$ have the following two properties:
(1) For all $x_1, x_2 \in \mathbb{R}$ such that $x_1 < x_2$, we have $$f \left( x_1 \right) \leq f \left( x_2 \right). $$
(2) For all $x_1, x_2 \in \mathbb{R}$, we have $$f \left( x_1 + x_2 \right) = f \left( x_1 \right) + f \left( x_2 \right). $$
Is such a function $f$ continuous at every point $c$ of $\mathbb{R}$?
My Effort:
We can show that for every rational number $q$, we have $$ f(q) = q f(1). $$
As $f$ is monotonic (increasing), so the set of points of discontinuity of $f$ is at most countable.
How to proceed from here?
If we could show that $f$ is continuous at every rational point $c \in \mathbb{Q}$, then $f$ would also be continuous at every point of $\mathbb{R}$. Am I right?
What next?
Context:
Sec. 5.6 (in fact immediately after Theorem 5.6.4) in the book Introduction To Real Analysis by Robert G. Bartle & Donald R. Sherbert, 4th edition.