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.
In class we were given 3 main ways to solve continuity proofs.
A function "f" is continuous at x=a if:
Limit of f(x) as x->a = f(a), (a can be all real numbers)
Limit of f(a+h) as h->0 = f(a), (Let x=a+h)
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.
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
Only way this can be possible is if f(0)=0
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!