# Cauhcy functional Equation

I am aware there are a variety of these questions in this website, however the variant of my question is very different so please do not consider closing my question.

As for the questions it says the following:

Let f be a lebesgue measurable function such that
$$f: (0,infty) to mathbb{R} \ f(x+y) = f(x) + f(y)$$ for all $$x,y geq 0$$. Fix some $$a in mathbb{R}$$, set $$Lambda_a subset (0,infty)$$, and let $$Lambda_a = { x geq 0: f(x) geq ax }$$.

1. Let $$mu$$ denote the Lebesgue measure. Show that if $$a in mathbb{R}$$ such that $$mu(Lambda_a) > 0$$, then $$Lambda_a$$ contains an interval of the form $$(b,infty)$$ for some positive $$b$$.

2. Show that in fact we may take $$b=0$$ in part 1.

3. Prove that there exists a $$lambda in mathbb{R}$$ such that $$f(x) = lambda x$$ for all $$x geq 0$$.