This is an exercise from a past exam I'm using to try to help me study. Let $f$ be measurable and bounded on $[0,1]$ satisfying $f(x+y)=f(x)+f(y);\quad f(1)=1.$ I'm trying to show that $f(x)=x$. We're given a hint to show it is continuous by using the hypothesis in a "mildly clever" way, show that it is the identity on the rational points, then extend by continuity.
I am able to show that the function is the identity on $[0,1]\cap\mathbb{Q}$ without any trouble, but I'm not sure how to show it's continuous. From there, showing the function is the identity would be easy since it follows from the fact that the rational numbers are dense in $\mathbb{R}$.