Nowhere Continuous Function Dirichlet Proof

[cmajor47]

Homework Statement

Prove that the Dirichlet function is continuous nowhere.

Homework Equations

Dirichlet function = 1 when x is rational, and 0 when x is irrational.

The Attempt at a Solution

I was looking at this proof on http://math.feld.cvut.cz/mt/txtd/1/txe4da1c.htm
At the very end when the creator shows that inf f(x) \neq sup f(x)
how does this tell you that the function is never continuous?
Do the greatest lower bound and least upper bound of a function have to be equal at some point for a function to be continuous?

 

[Hurkyl]

At the very end when the creator shows that inf f(x) \neq sup f(x)

That's not what he showed.

He showed the infimum (over all P) of U(f,P) was not the supremum (over all P) of L(f,P).

And he wasn't proving anything about continuity anyways; that article is about Riemann integrability.

 

[cmajor47]

Oh, wow, don't know how I missed that. Thanks!