## Dirichlet Function

Let and be Real Numbers (usually taken as and ). The Dirichlet function is defined by

The function is Continuous at Irrational and discontinuous at Rational points. The function can be written analytically as

Because the Dirichlet function cannot be plotted without producing a solid blend of lines, a modified version can be defined as

(Dixon 1991), illustrated above.

Tall, D. The Gradient of a Graph.'' Math. Teaching 111, 48-52, 1985.