Density of the irrational numbers ($\overline{\mathbb{R}\setminus \mathbb{Q}}=\mathbb{R}$)

Every nonempty open real interval contains an irrational number. Choose a rational number $q$ strictly inside the interval using density of the rational numbers; then $q+\sqrt{2}/N$ remains inside for sufficiently large positive integers $N$ and is irrational. Thus the irrational numbers form a dense subset of the real line. Together with rational density, this gives approaching sequences of both types at every real point, useful for testing continuity of functions defined differently on the two sets.