1. The problem statement, all variables and given/known data Prove that the Dirichlet function is continuous nowhere. 2. Relevant equations Dirichlet function = 1 when x is rational, and 0 when x is irrational. 3. 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?