I have read on some websites that if f: R -> R is continous for every x in R, then f(x+y) = f(x) + f(y) defines f as a linear function. Now, I am given: Code (Text): Suppose f is continuous at 0, and that for all x, y in R, f(x+y) = f(x) + f(y). a) Show that f(0) = 0. b) Prove that f is continuous at every point a in R. Solution for a) Code (Text): f(0+0) = f(0) + f(0) f(0) = 2f(0) 0 = 2f(0) - f(0) 0 = f(0) I am confused as to how to go about with part b)? (note: this question was under the topc of limits and continuity.) So I was planning to use limits as part of the solution to part b). Thanks in Advance In fact, there are series of questions following this, that is in a similar format, but with f(x+y) = f(x)f(y). And once again, continuity of the function must be proved. If you guys can help me with the first one, I will try to do the second one by myself, but if there are any tips or tricks involved in the second proof, please hint me. Thank you.