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## Homework Statement

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.

## Homework 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.

## 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!