- #1

cmajor47

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## 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) [tex]\neq[/tex] 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?