Prove Continuity of f at a w/ f(x+y)=f(x)+f(y)

[freshlikeuhh]

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

f(x+y) = f(x) + f(y)
Limit Definition
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.

 

[javierR]

It may help to state what you wish to show explicitly. Based on the definition of continuity, you want to show that lim_{x-->a}[f(x)]=f(a) (at the same time you'll verify that the limit exists). Can you see how to redefine variables and rewrite the expression above so that this expression is what the book solution is verifying (though the last step before finding "0" seems superfluous)?

 

[freshlikeuhh]

It may help to state what you wish to show explicitly. Based on the definition of continuity, you want to show that lim_{x-->a}[f(x)]=f(a) (at the same time you'll verify that the limit exists). Can you see how to redefine variables and rewrite the expression above so that this expression is what the book solution is verifying (though the last step before finding "0" seems superfluous)?

Well, I understand how the property f(x+y)=f(x)+f(y) is used to demonstrate f(0)=0 and obviously it is employed to expand f(a+h)=f(a)+f(h). I understand that this step is supposed to demonstrate the continuity of all a, but then why is h->0, and how exactly is continuity of a demonstrated if a is canceled out? Is this just using that fact that f(a - a) = f(0), which is 0?

What would an alternate solution look like?