### 8.12.05

## nowhere continuous bijection of the unit interval onto itself

if x is rational: f(x) = x

if x is irrational and x < 1/2: f(x) = x + 1/2

if x is irrational and x ≥ 1/2: f(x) = x - 1/2

given any rational number x, there is an irrational number y as close to it as you like with f(y) about 1/2 away from f(x).

similarly, given any irrational number we can find a rational number as close to it as we like, and their images under f will be about 1/2 apart.

if x is irrational and x < 1/2: f(x) = x + 1/2

if x is irrational and x ≥ 1/2: f(x) = x - 1/2

given any rational number x, there is an irrational number y as close to it as you like with f(y) about 1/2 away from f(x).

similarly, given any irrational number we can find a rational number as close to it as we like, and their images under f will be about 1/2 apart.