TITLE: If $f$ is continuous and $f(x+y) = f(x)+f(y)$, then $f(x) = cx$ for all $x \in \mathbb{R}$
QUESTION [5 upvotes]: Assume $f$ is a function over $\mathbb{R}$ satisfying $f(x+y) = f(x)+f(y)$ for all $x,y \in \mathbb{R}$. Show that if $f$ is continuous, then $f(x) = cx$ for all $x \in \mathbb{R}.$

I find it hard to use the definition of continuity to derive anything from the equation. I think we could write the definition of the limit in terms to get $$\forall \epsilon,\exists \delta \quad 0<|x-a|<\delta \quad \implies \quad |f(x)-L| < \epsilon.$$
I am not sure where to go from here.

REPLY [2 votes]: Hint: Check that the claim holds for integers. Let $f(1) = \alpha$. Then use the relation to show that $f(m) = m \alpha$ for all integers. 
Show that in fact the above result holds for rational numbers.
Use the fact that continuity in a metric space is equivalent to sequential continuity. 
Use that every real number is a limit of a sequence of rational numbers together with continuity to conclude the result.