Show if $f(x+y)=f(x)+f(y)$ for all $x,y$ and $f$ is Lebesgue measurable, then $f$ is continuous.
Let $g:\mathbb{R}\to\mathbb{R}$ be a measurable function such that $g(x+y) =g(x)+g(y).$ Then $g(x) = g(1)x$ .