I was looking at a post on MathOverflow about "What is your favorite 'strange' function?" One of the answers mentioned Thomae's function claiming that the function was "continuous at all irrationals and discontinuous at all rationals."
This got me to thinking, how are irrational and rational numbers related on the real number line? Is every irrational number "surrounded" by two rational number "neighbors?" For instance, can we think of the non-negative real number line as:
Zero (rational), an irrational number infinitesimally close to zero, a rational number infinitesimally larger than the previous irrational number, an irrational number infinitesimally larger than the previous rational number, etc. ?