# Homework Help: Prove function continuous at only one point

1. Nov 30, 2009

### cmajor47

1. The problem statement, all variables and given/known data
Prove that the function defined as f(x)= x when x is rational and -x when x is irrational is only continuous at 0.

2. Relevant equations

3. The attempt at a solution
I have been looking at this website which proves this:
http://planetmath.org/encyclopedia/FunctionContinuousAtOnlyOnePoint.html [Broken]

I don't understand how the writer proves that the function is not continuous when a$$\neq$$0.
Why do they take xk= 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.

Last edited by a moderator: May 4, 2017
2. Nov 30, 2009

### CompuChip

Do you agree that
"f is continuous at a, if for all sequences xk that converge to a, f(xk) 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(xk) 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.