1. The problem statement, all variables and given/known data 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. 2. Relevant equations In class we were given 3 main ways to solve continuity proofs. A function "f" is continuous at x=a if: a.) Limit of f(x) as x->a = f(a), (a can be all real numbers) b.) Limit of f(a+h) as h->0 = f(a), (Let x=a+h) c.) (delta-epsilon proof) For all epsilon greater than 0, there exists some delta greater than 0, such that for all x, if |x-a|< delta then |f(x)-f(a)|< epsilon. 3. The attempt at a solution So far, by working with my teacher I was able to get this much as being correct: Theres not enough information for a delta-epsilon proof so [ Were told that f(x+y)=f(x)+f(y), for all x,y belonging to real numbers then In partiality: f(0+0)=f(0)+f(0) f(0)=2f(0) Only way this can be possible is if f(0)=0 hence f(0)=0 Now consider: Limit of f(a+h)-f(a) as h->0 is equal to 0. ] Im not as to what i should do now and what the second part that my teacher wants me to consider will prove. Any ideas would be greatly appreciated!