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Prove the Dirichlet function does not have a limit? - Yahoo! Answers
If $\displaystyle \alpha>0$ then there is some $\displaystyle t\in (-\alpha,\alpha)$ which is rational and there is an $\displaystyle s\in (-\alpha,\alpha)$ which is irrational. That is, in any open interval there is a rational and an irrational number. So there are two places 16 units apart. Continuous functions must have values 'close together' at points of continuity.