Let $f:\Bbb R\to \Bbb R$ be additive i.e. satisfies $f(x+y)=f(x)+f(y)$ for all $x$, $y$. and Lebesgue measurable.
Show that $f$ is continuous and hence of the form $f(x)=cx$.
In order to show that $f$ is continuous at a point $a$ we have to show that $|f(x)-f(a)|<\epsilon $ whenever $|x-a|<\delta$.
But I can't find any way to proceed here. Please give some hints.