MT2002 Analysis

 Previous page (The epsilon-delta definition) Contents Next page (Metric spaces: definition and examples)

## 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 pQ and let (xn) be a sequence of irrationals converging to p. Then f (p) = 1 but f (xn))→ 0 and so f is discontinuous at p.
Similarly, if pQ 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/bQ and let (xn) be a sequence of irrationals converging to p. Then f (p) = 1/b ≠ the limit of f (xn)).
However, if pQ 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 (xn))→ 0 = f (p) and so f is continuous at these points.

 Previous page (The epsilon-delta definition) Contents Next page (Metric spaces: definition and examples)

JOC September 2001