f satisfies that for all x,y f(x+y) = f(x) + f(y)
show that f is continuous on R
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?