Prove function continuous at only one point

[cmajor47]

Homework Statement

Prove that the function defined as f(x)= x when x is rational and -x when x is irrational is only continuous at 0.

Homework Equations

The Attempt at a Solution

I have been looking at this website which proves this:
http://planetmath.org/encyclopedia/FunctionContinuousAtOnlyOnePoint.html

I don't understand how the writer proves that the function is not continuous when a$$\neq$$0.
Why do they take x_k= to something? How does this help them show that the limits equal a and -a.
How does showing that prove that f is not continuous at a?

I know those are a lot of question, but maybe an answer to even just the first ones would help me understand this proof.

 

[CompuChip]

Do you agree that
"f is continuous at a, if for all sequences x_k that converge to a, f(x_k) converges to f(a)" ?

So then to show that f is not continuous at a, all you need is to find one sequence that does converge to a but for which the limit of the f(x_k) is not f(a). (The negation of "for all sequences S, P(S)" is "there exists a sequence S such that not P(S)").

So what they do is take two different sequences which both have limit a, and show that when you apply f first and take the limit, you get two different results. Since f(a) cannot be equal to a and -a at the same time, at least one of this sequence forms a counter-example to the statement of continuity they start out with.