Dirichlet Function

Let $c$ and $d\not=c$ be Real Numbers (usually taken as $c=1$ and $d=0$). The Dirichlet function is defined by

$$D(x)=\cases{ c & for $x$\ rational\cr d & for $x$\ irrational.}$$

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

$$D(x)=\lim_{m, n\to\infty} \cos[(m!\pi x)^n].$$

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

$$D_M(x)=\cases{ 0 & for $x$\ irrational\cr b & for $x=a/b$\ with $a/b$\ a reduced fraction\cr}$$

(Dixon 1991), illustrated above.

See also Continuous Function, Irrational Number, Rational Number

References

Dixon, R. Mathographics. New York: Dover, pp. 177 and 184-186, 1991.

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