Suppose that f satisfies f(x+y) = f(x) + f(y), and that f is continuous at 0. Prove that f is continuous at a for all a.
f(x+y) = f(x) + f(y)
Continuity: f is continuous at a if the limit as x approaches a is the value of the function at a.
The Attempt at a Solution
I am not even sure how to approach this question. I have already seen a solution to this question but I do not understand it. That solution is as follows:
Note that f(x+0) = f(x) + f(0), so f(0)=0. Now:
(h->0)lim f(a+h) - f(a) = (h->0)lim f(a) + f(h) - f(a) = (h->0)lim f(h) = (h->0) f(h) - f(0) = 0, since f is continuous at 0.
This is in the back of my textbook and I don't understand how that concludes anything.