Some horrible functions

Here are a couple of the functions which originally forced mathematicians to refine their ideas of continuity.

1. Let f be the function defined by f (x) = 1 if x is rational and f (x) = 0 if x is irrational.
   Then f is discontinuous at every point x.

   Proof
   Take p ∈ Q and let (x_n) be a sequence of irrationals converging to p. Then f (p) = 1 but f (x_n))→ 0 and so f is discontinuous at p.
   Similarly, if p ∉ Q then choose a sequence of rationals converging to p and deduce the same result.
   
   An even cleverer (and more horrible) function is the following amazing example due to Dirichlet (1805 to 1859).

2. Let f be the function defined by f (x) = 0 if x is irrational and f (x) = ¹/_b if x is the rational number ^a/_b (in lowest terms).
   Then f is discontinuous at every rational point, but continuous at every irrational point.

   Proof
   Take p = ^a/_b ∈ Q and let (x_n) be a sequence of irrationals converging to p. Then f (p) = ¹/_b ≠ the limit of f (x_n)).
   However, if p ∉ Q then given ε > 0, we may find an interval around p which misses all the rationals of the form ^a/_b with ¹/_b < ε. Then for sequences lying in this interval we do have (f (x_n))→ 0 = f (p) and so f is continuous at these points.