1. The problem statement, all variables and given/known data f:R->R f satisfies that for all x,y f(x+y) = f(x) + f(y) show that f is continuous on R 3. The attempt at a solution I assumed that limit of f at 0 existed. then i showed that that limit must be zero and that f(0)=0, so f is continuous at 0. From there, i broke it up into different cases (integer, inverse of an integer, rational number, irrational number) and showed that f must be continuous at each of them. My question is whether or not my assumption that limit of f at 0 existed is a valid assumption to make. Is it even necessarily true? if so how would i go about proving it?